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[http://balthier.ecn.purdue.edu/index.php/ECE662 ECE662 Main Page]
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=Lecture 17, [[ECE662]]: Decision Theory=
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Lecture notes for [[ECE662:BoutinSpring08_Old_Kiwi|ECE662 Spring 2008]], Prof. [[user:mboutin|Boutin]].
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[http://balthier.ecn.purdue.edu/index.php/ECE662#Class_Lecture_Notes Class Lecture Notes]
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Other lectures: [[Lecture 1 - Introduction_Old Kiwi|1]],
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[[Lecture 2 - Decision Hypersurfaces_Old Kiwi|2]],
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[[Lecture 3 - Bayes classification_Old Kiwi|3]],
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[[Lecture 4 - Bayes Classification_Old Kiwi|4]],
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[[Lecture 5 - Discriminant Functions_Old Kiwi|5]],
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[[Lecture 6 - Discriminant Functions_Old Kiwi|6]],
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[[Lecture 7 - MLE and BPE_Old Kiwi|7]],
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[[Lecture 8 - MLE, BPE and Linear Discriminant Functions_Old Kiwi|8]],
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[[Lecture 9 - Linear Discriminant Functions_Old Kiwi|9]],
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[[Lecture 10 - Batch Perceptron and Fisher Linear Discriminant_Old Kiwi|10]],
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[[Lecture 11 - Fischer's Linear Discriminant again_Old Kiwi|11]],
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[[Lecture 12 - Support Vector Machine and Quadratic Optimization Problem_Old Kiwi|12]],
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[[Lecture 13 - Kernel function for SVMs and ANNs introduction_Old Kiwi|13]], 
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[[Lecture 14 - ANNs, Non-parametric Density Estimation (Parzen Window)_Old Kiwi|14]],
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[[Lecture 15 - Parzen Window Method_Old Kiwi|15]],
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[[Lecture 16 - Parzen Window Method and K-nearest Neighbor Density Estimate_Old Kiwi|16]],
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[[Lecture 17 - Nearest Neighbors Clarification Rule and Metrics_Old Kiwi|17]],
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[[Lecture 18 - Nearest Neighbors Clarification Rule and Metrics(Continued)_Old Kiwi|18]],
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[[Lecture 19 - Nearest Neighbor Error Rates_Old Kiwi|19]],
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[[Lecture 20 - Density Estimation using Series Expansion and Decision Trees_Old Kiwi|20]],
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[[Lecture 21 - Decision Trees(Continued)_Old Kiwi|21]],
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[[Lecture 22 - Decision Trees and Clustering_Old Kiwi|22]],
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[[Lecture 23 - Spanning Trees_Old Kiwi|23]],
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[[Lecture 24 - Clustering and Hierarchical Clustering_Old Kiwi|24]],
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[[Lecture 25 - Clustering Algorithms_Old Kiwi|25]],
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[[Lecture 26 - Statistical Clustering Methods_Old Kiwi|26]],
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[[Lecture 27 - Clustering by finding valleys of densities_Old Kiwi|27]],
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[[Lecture 28 - Final lecture_Old Kiwi|28]],
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----
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----
 
'''Nearest Neighbor Classification Rule'''
 
'''Nearest Neighbor Classification Rule'''
 
* useful when there are several labels
 
* useful when there are several labels
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* Reflexivity <math>D(\vec{x},\vec{x})=0, \forall \vec{x}\in S</math>
 
* Reflexivity <math>D(\vec{x},\vec{x})=0, \forall \vec{x}\in S</math>
 
* Triangle Inequality <math>D(\vec{x_1},\vec{x_2})+D(\vec{x_2},\vec{x_3})\geq D(\vec{x_1},\vec{x_3}) , \forall \vec{x_1}, \vec{x_2}, \vec{x_3}\in S</math>
 
* Triangle Inequality <math>D(\vec{x_1},\vec{x_2})+D(\vec{x_2},\vec{x_3})\geq D(\vec{x_1},\vec{x_3}) , \forall \vec{x_1}, \vec{x_2}, \vec{x_3}\in S</math>
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[[Image:distances_Old Kiwi.jpg]]
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'''Illustration of 3 different metrics'''
  
  
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Euclidean distance: <math>D(\vec{x_1},\vec{x_2})=||\vec{x_1}-\vec{x_2}||_{L_2}=\sqrt{\sum_{i=1}^n ({x_1}^i-{x_2}^i)^2}</math>
 
Euclidean distance: <math>D(\vec{x_1},\vec{x_2})=||\vec{x_1}-\vec{x_2}||_{L_2}=\sqrt{\sum_{i=1}^n ({x_1}^i-{x_2}^i)^2}</math>
  
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Manhattan distance: <math>D(\vec{x_1},\vec{x_2})=||\vec{x_1}-\vec{x_2}||_{L_1}=\sum_{i=1}^n |{x_1}^i-{x_2}^i|</math>
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Manhattan (cab driver) distance: <math>D(\vec{x_1},\vec{x_2})=||\vec{x_1}-\vec{x_2}||_{L_1}=\sum_{i=1}^n |{x_1}^i-{x_2}^i|</math>
  
 
Minkowski metric: <math>D(\vec{x_1},\vec{x_2})=||\vec{x_1}-\vec{x_2}||_{L_p}=(\sum_{i=1}^n ({x_1}^i-{x_2}^i)^p)^{\frac{1}{p}}</math>
 
Minkowski metric: <math>D(\vec{x_1},\vec{x_2})=||\vec{x_1}-\vec{x_2}||_{L_p}=(\sum_{i=1}^n ({x_1}^i-{x_2}^i)^p)^{\frac{1}{p}}</math>
  
 
Riemannian metric: <math>D(\vec{x_1},\vec{x_2})=\sqrt{(\vec{x_1}-\vec{x_2})^\top \mathbb{M}(\vec{x_1}-\vec{x_2})}</math>
 
Riemannian metric: <math>D(\vec{x_1},\vec{x_2})=\sqrt{(\vec{x_1}-\vec{x_2})^\top \mathbb{M}(\vec{x_1}-\vec{x_2})}</math>
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Infinite norm: <math>D(\vec{x_1},\vec{x_2})=||\vec{x_1}-\vec{x_2}||_{\infty}=max_i |{x_1}^i-{x_2}^i|</math>
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where M is a symmetric positive definite <math>n\times n</math> matrix. Different choices for M enable associating different weights with different components.
 
where M is a symmetric positive definite <math>n\times n</math> matrix. Different choices for M enable associating different weights with different components.
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<math>x_2</math>={fever, neckstiffness}
 
<math>x_2</math>={fever, neckstiffness}
  
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Tanimoto metric
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<math>D(set1, set2) = \frac {|set1|+|set2|-2|set1 \bigcap set2| }{|set1|+|set2|-|set1 \bigcap set2|} </math>
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Example: previous approach to shape recognition
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Given is a set of ordered points in <math>R_n =(p_1,p_2,\cdots,p_N)</math>
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We want to recognize the shape
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[[Image:Lec17_fig1_Old Kiwi.JPG]]
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Figure 1
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Given template (triangle form): (T1,T2,...,TN);
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We want to assign one of test template to a test (P1,P2,P3)
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In this case, we should not use Euclidean distance!,
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[[Image:Lec17_fig2_Old Kiwi.JPG]]
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Figure 2
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becasue shape defined by point is unchanged (invariant) by rotation and translation of triangles.
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Therefore, distance between 2 triangles (or shapes) must be independent on the position and orientation of triangles.
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'''Procrustes metric'''
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<math>D(p,\bar p)=  \sum_{\begin{matrix}i=1 \\ rotation R, translation T \end{matrix}}^n
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{\begin{Vmatrix} Rp_i+T-\bar p_i \end{Vmatrix}} _{L^2} </math>
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<math>p=(p_1, p_2, \cdots ,p_N),\bar p = (\bar p_1, \bar p_2, \cdots ,\bar p_N)</math>
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Alternative approach
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"Use invariant coordinate to repeat <math>p=(p_1, p_2, \cdots ,p_N) </math> "
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i.e find <math> \varphi </math> such that
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<math> \varphi : \mathbb{R}^n\rightarrow \mathbb{R}^k</math>
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(where, typically <math> k \leq n </math>)
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s.t <math> \varphi  (x) = \varphi (\bar x) </math>
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whenever <math> \exists </math> R, T with <math> R \bar X + \bar T = X</math>
  
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Example of phi with triangle (Figure 1):
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Example of phi with triangle (Figure 3):
 
[[Image:lec17_rot_tri_Old Kiwi.png]]
 
[[Image:lec17_rot_tri_Old Kiwi.png]]
  
 
(p1,p2,p3) -> (new p1, new p2, new p3)
 
(p1,p2,p3) -> (new p1, new p2, new p3)
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[[ECE662:BoutinSpring08_Old_Kiwi|Back to ECE662, Spring 2008, Prof. Boutin]]
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[[Category:Lecture Notes]]

Latest revision as of 08:38, 17 January 2013

Lecture 17, ECE662: Decision Theory

Lecture notes for ECE662 Spring 2008, Prof. Boutin.

Other lectures: 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,

Nearest Neighbor Classification Rule

  • useful when there are several labels
  • e.g. fingerprint-based recognition

Problem: Given the labeled training samples: $ \vec{X_1}, \vec{X_2}, \ldots, \vec{X_d} $ $ \in \mathbb{R}^n $ (or some other feature space) and an unlabeled test point $ \vec{X_0} $ $ \in \mathbb{R}^n $.

Classification: Let $ \vec{X_i} $ be the closest training point to $ \vec{X_0} $, then we assign the class of $ \vec{X_i} $ to $ \vec{X_0} $.


What do we mean by closest?

There are many meaning depending on the metric we choose for the feature space.


Definition A "metric" on a space S is a function

$ D: S\times S\rightarrow \mathbb{R} $

that satisfies the following 4 properties:

  • Non-negativity $ D(\vec{x_1},\vec{x_2})\geq 0, \forall \vec{x_1},\vec{x_2}\in S $
  • Symmetry $ D(\vec{x_1},\vec{x_2})=D(\vec{x_2},\vec{x_1}), \forall \vec{x_1},\vec{x_2}\in S $
  • Reflexivity $ D(\vec{x},\vec{x})=0, \forall \vec{x}\in S $
  • Triangle Inequality $ D(\vec{x_1},\vec{x_2})+D(\vec{x_2},\vec{x_3})\geq D(\vec{x_1},\vec{x_3}) , \forall \vec{x_1}, \vec{x_2}, \vec{x_3}\in S $

Distances Old Kiwi.jpg Illustration of 3 different metrics


Examples of metrics

Euclidean distance: $ D(\vec{x_1},\vec{x_2})=||\vec{x_1}-\vec{x_2}||_{L_2}=\sqrt{\sum_{i=1}^n ({x_1}^i-{x_2}^i)^2} $

Manhattan (cab driver) distance: $ D(\vec{x_1},\vec{x_2})=||\vec{x_1}-\vec{x_2}||_{L_1}=\sum_{i=1}^n |{x_1}^i-{x_2}^i| $

Minkowski metric: $ D(\vec{x_1},\vec{x_2})=||\vec{x_1}-\vec{x_2}||_{L_p}=(\sum_{i=1}^n ({x_1}^i-{x_2}^i)^p)^{\frac{1}{p}} $

Riemannian metric: $ D(\vec{x_1},\vec{x_2})=\sqrt{(\vec{x_1}-\vec{x_2})^\top \mathbb{M}(\vec{x_1}-\vec{x_2})} $

Infinite norm: $ D(\vec{x_1},\vec{x_2})=||\vec{x_1}-\vec{x_2}||_{\infty}=max_i |{x_1}^i-{x_2}^i| $


where M is a symmetric positive definite $ n\times n $ matrix. Different choices for M enable associating different weights with different components.

In this way, we see that $ \mathbb{R}^n $, $ \mathbb{Z}^n $, $ \mathbb{C}^n $ have many natural metrics, but feature could be in some other set, e.g. a discrete set.

for example,

$ x_1 $={fever, skinrash, high blodd pressure}

$ x_2 $={fever, neckstiffness}

Tanimoto metric

$ D(set1, set2) = \frac {|set1|+|set2|-2|set1 \bigcap set2| }{|set1|+|set2|-|set1 \bigcap set2|} $

Example: previous approach to shape recognition Given is a set of ordered points in $ R_n =(p_1,p_2,\cdots,p_N) $ We want to recognize the shape

Lec17 fig1 Old Kiwi.JPG Figure 1

Given template (triangle form): (T1,T2,...,TN); We want to assign one of test template to a test (P1,P2,P3) In this case, we should not use Euclidean distance!,

Lec17 fig2 Old Kiwi.JPG Figure 2

becasue shape defined by point is unchanged (invariant) by rotation and translation of triangles.

Therefore, distance between 2 triangles (or shapes) must be independent on the position and orientation of triangles.

Procrustes metric

$ D(p,\bar p)= \sum_{\begin{matrix}i=1 \\ rotation R, translation T \end{matrix}}^n {\begin{Vmatrix} Rp_i+T-\bar p_i \end{Vmatrix}} _{L^2} $

$ p=(p_1, p_2, \cdots ,p_N),\bar p = (\bar p_1, \bar p_2, \cdots ,\bar p_N) $

Alternative approach "Use invariant coordinate to repeat $ p=(p_1, p_2, \cdots ,p_N) $ "

i.e find $ \varphi $ such that

$ \varphi : \mathbb{R}^n\rightarrow \mathbb{R}^k $ (where, typically $ k \leq n $)

s.t $ \varphi (x) = \varphi (\bar x) $

whenever $ \exists $ R, T with $ R \bar X + \bar T = X $

Example of phi with triangle (Figure 3): Lec17 rot tri Old Kiwi.png

(p1,p2,p3) -> (new p1, new p2, new p3)

Back to ECE662, Spring 2008, Prof. Boutin

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