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[http://balthier.ecn.purdue.edu/index.php/ECE662#Class_Lecture_Notes Class Lecture Notes]
 
 
 
== Lecture Theme ==
 
== Lecture Theme ==
  
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Next: [[Lecture 3_OldKiwi]]
 
Next: [[Lecture 3_OldKiwi]]
  
== Lectures ==
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[[Category:Lecture Notes]]
[http://balthier.ecn.purdue.edu/index.php/Lecture_1_-_Introduction 1] [http://balthier.ecn.purdue.edu/index.php/Lecture_2_-_Decision_Hypersurfaces 2] [http://balthier.ecn.purdue.edu/index.php/Lecture_3_-_Bayes_classification 3]
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[http://balthier.ecn.purdue.edu/index.php/Lecture_4_-_Bayes_Classification 4] [http://balthier.ecn.purdue.edu/index.php/Lecture_5_-_Discriminant_Functions 5] [http://balthier.ecn.purdue.edu/index.php/Lecture_6_-_Discriminant_Functions 6] [http://balthier.ecn.purdue.edu/index.php/Lecture_7_-_MLE_and_BPE 7] [http://balthier.ecn.purdue.edu/index.php/Lecture_8_-_MLE%2C_BPE_and_Linear_Discriminant_Functions 8] [http://balthier.ecn.purdue.edu/index.php/Lecture_9_-_Linear_Discriminant_Functions 9] [http://balthier.ecn.purdue.edu/index.php/Lecture_10_-_Batch_Perceptron_and_Fisher_Linear_Discriminant 10] [http://balthier.ecn.purdue.edu/index.php/Lecture_11_-_Fischer%27s_Linear_Discriminant_again 11] [http://balthier.ecn.purdue.edu/index.php/Lecture_12_-_Support_Vector_Machine_and_Quadratic_Optimization_Problem 12] [http://balthier.ecn.purdue.edu/index.php/Lecture_13_-_Kernel_function_for_SVMs_and_ANNs_introduction 13] [http://balthier.ecn.purdue.edu/index.php/Lecture_14_-_ANNs%2C_Non-parametric_Density_Estimation_%28Parzen_Window%29 14] [http://balthier.ecn.purdue.edu/index.php/Lecture_15_-_Parzen_Window_Method 15] [http://balthier.ecn.purdue.edu/index.php/Lecture_16_-_Parzen_Window_Method_and_K-nearest_Neighbor_Density_Estimate 16] [http://balthier.ecn.purdue.edu/index.php/Lecture_17_-_Nearest_Neighbors_Clarification_Rule_and_Metrics 17] [http://balthier.ecn.purdue.edu/index.php/Lecture_18_-_Nearest_Neighbors_Clarification_Rule_and_Metrics%28Continued%29 18]
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[http://balthier.ecn.purdue.edu/index.php/Lecture_19_-_Nearest_Neighbor_Error_Rates 19]
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[http://balthier.ecn.purdue.edu/index.php/Lecture_20_-_Density_Estimation_using_Series_Expansion_and_Decision_Trees 20]
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Revision as of 22:13, 6 April 2008

Lecture Theme

Decision hypersurfaces

Actual Lecture

Making decisions is the art of drawing hypersurfaces.

Hypersurfaces_OldKiwi discussed in this lecture

These Hypersurfaces_OldKiwi are discussed in this lecture:

  • Linear Separations
    • Straight lines in 2D $ ax+by+c=0 $
    • Planes in 3D $ ax+by+cx+d=0 $
    • Hyperplanes in nD $ \sum a_i x_i = \textrm{constant} $
  • Union of linear separations
  • Tree-based separation

Combining Hypersurfaces

Main article: Combining Hypersurfaces_OldKiwi

Hypersurfaces are combined by multiplication of the functions which define them, not by intersection.

Tree Based Separation

By using union of hypersurfaces, classes can be separated by a tree based structure. Tree based separation is harder, and obtaining an optimum decision tree is another difficult problem in pattern recognition.

Example:

Treebased OldKiwi.jpg


Question: What do we mean by good separation?

Question: What do we mean by good separation?


Meaning 1: One that will make few mistakes when used to classify actual data. In statistical language: small probability of error.

Meaning 2: One that will make mistakes that will not be too costly. In statistical language: small expected cost of errors.

Note

If the cost function $ c(x) $ from Meaning 2 is defined to be 1 for an error, and 0 otherwise, then $ E[c(x)] = P[error] $, so that Meaning 1 is a special case of Meaning 2.

Illustration

Take 50,000 people in an airport. We want to be able to distinguish men from women among these 50,000 people.

Use hairlength as the single feature.

File:Hairlength.jpg

Histogram represents the actual numbers.

Decision hypersurface: $ l-6=0 $. This is the best decision rule in the sense that it minimizes the probability of error. Can not do better than this for the scenario where only hairlength is used as the feature.

Remember

  • Good features produce distinct bumps. That is try to choose features that have a natural grouping for a particular class. For example, hair-length is a good feature to separate men from women as more most men are likely to have short hair-length.
  • Bad features (e.g. background color) produce overlapping bumps.
  • In order to get a good decision algorithm, we need to have good features (and good separation too).

Work in low dimensionality as far as possible

Why? This is typically explained as the curse of dimensionality_OldKiwi. For more, this term can be googled. There is a nice paper in VLDB 2000 that discusses the notion of nearest-neighbors in high dimensional spaces.

Let $ x_{1},x_{2},\ldots,x_{N} $ be features. Suppose we use a large number of features to come up with a decision hypersurface $ g(x_{1},x_{2},\ldots,x_{N})=0 $

Then g is the single feature we need.

Previous: Lecture 1-Introduction_OldKiwi Next: Lecture 3_OldKiwi

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