Contents
Lecture Theme
Decision hypersurfaces
Actual Lecture
Making decisions is the art of drawing hypersurfaces.
Hypersurfaces_Old Kiwi discussed in this lecture
These Hypersurfaces_Old Kiwi are discussed in this lecture:
- Linear Separations
- Straight lines in 2D |line|
- Planes in 3D |plane|
- Hyperplanes in nD |hyperplane|
- Union of linear separations
- Tree-based separation
.. |line| image:: tex
- alt: tex:ax+by+c=0
.. |plane| image:: tex
- alt: tex:ax+by+cz+d=0
.. |hyperplane| image:: tex
- alt: tex:\sum a_{i}x_{i}=const.
Combining Hypersurfaces
Main article: Combining Hypersurfaces_Old Kiwi
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:
.. image:: Treebased.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 |cdef| from Meaning 2 is defined to be 1 for an error, and 0 otherwise, then |Ecost|, so that Meaning 1 is a special case of Meaning 2.
.. |cdef| image:: tex
- alt: tex:c(x)
.. |Ecost| image:: tex
- alt: tex:E[c(x)] = P[error]
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.
.. image:: hairlength.jpg
Histogram represents the actual numbers.
Decision hypersurface: |hair hyp|. 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.
.. |hair hyp| image:: tex
- alt: tex:l-6=0
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_Old Kiwi. 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 (See citeseer)
Let |features| be features. Suppose we use a large number of features to come up with a decision hypersurface |g|
.. |features| image:: tex
- alt: tex:x_{1},x_{2},\ldots,x_{N}
.. |g| image:: tex
- alt: tex:g(x_{1},x_{2},\ldots,x_{N})=0
Then g is the single feature we need.
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