Line 46: | Line 46: | ||
* Nearest Neighbor error rate | * Nearest Neighbor error rate | ||
recall | recall | ||
− | Probability of error (error rate) on test data is | + | Probability of error (error rate) on test data is |
<math>P(e)=\int p(e \mid \vec{x}) p(\vec{x} d\vec{x})</math> | <math>P(e)=\int p(e \mid \vec{x}) p(\vec{x} d\vec{x})</math> | ||
Let <math>P_d(e)</math> be the error rate when d training samples are used. | Let <math>P_d(e)</math> be the error rate when d training samples are used. | ||
− | Let <math>P=\lim_{d \rightarrow \infty } P_{d}(e) </math> | + | |
+ | Let <math>P=\lim_{d \rightarrow \infty } P_{d}(e) </math> | ||
Claim | Claim | ||
limit error rate <math>P=\int (1-\sum _{i=1} ^{c}p^2 (\omega _i \mid \vec{x}))p(x)dx</math> | limit error rate <math>P=\int (1-\sum _{i=1} ^{c}p^2 (\omega _i \mid \vec{x}))p(x)dx</math> | ||
− | Proof of claim: Given observation <math>\vec{x}</math>, denode by <math>\vec{x}_d</math> the | + | Proof of claim: Given observation <math>\vec{x}</math>, denode by <math>\vec{x}_d</math> the nearest neighbor of <math>\vec{x}</math> among <math>\{\vec{x}_1,\vec{x}_2, \cdots , \vec{x}_d \}</math> |
+ | |||
+ | <math>P_d(e \mid \vec{x})=\int p_d(e \mid \vec{x}, vec{x}_d)p_d(\vec{x}_d ^\' \mid \vec{x})</math> |
Revision as of 11:12, 10 March 2008
Nearest Neighbors Clarification Rule(Alternative Approaches) --Han47 10:34, 10 March 2008 (EDT)
Alternative Approach
find invariant coordination $ \varphi : \Re ^k \rightarrow \Re ^n $ --Han47 10:41, 10 March 2008 (EDT) such that $ \varphi (x) = \varphi (\bar x) $ for all $ x, \bar x $ which are related by a rotation & translation
Do not trivialize!
e.g.) $ \varphi (x) =0 $ gives us invariant coordinate but lose separation
Want $ \varphi (x) = \varphi (\bar x) $ $ \Leftrightarrow x, \bar x $ are related by a rotation and translation
Example $ p=(p_1,p_2,\cdots, p_N) \in \Re ^{3 \times N} $ $ \varphi $ maps representation position of taps on body onto $ (d_{12},d_{13},d_{14},\cdots , d_{N-1, N} ) $
where $ d_{ij} $= Euclidean distance between $ p_i $ and $ p_j $
Can reconstruct up to a rotation and translation
Warning: Euclidean distance in invariant coordination space has nothing to do with Euclidean distance or proanstes distance in initial feature space
Nearest Neighbor in $ \Re ^2 $ yields tessalation (tiling of floor with 2D shapes such that 1) no holes and 2) cover all of $ \Re ^2 $)
Shape of cells depends on metric chosen
E.g., if feature vectors are such that vectors related by a rotation belong to same class $ \rightarrow $ metric should be chosen so that files are rotationally symmetric.
Instead of working with (x,y) rotationally invariant, work with $ z=\sqrt(x^2 + y^2) $
How good is Nearest Neighbor rule? 1) training error is zero: don't care 2) test error? Want to Bayes error rate
- Nearest Neighbor error rate
recall Probability of error (error rate) on test data is $ P(e)=\int p(e \mid \vec{x}) p(\vec{x} d\vec{x}) $
Let $ P_d(e) $ be the error rate when d training samples are used.
Let $ P=\lim_{d \rightarrow \infty } P_{d}(e) $
Claim limit error rate $ P=\int (1-\sum _{i=1} ^{c}p^2 (\omega _i \mid \vec{x}))p(x)dx $
Proof of claim: Given observation $ \vec{x} $, denode by $ \vec{x}_d $ the nearest neighbor of $ \vec{x} $ among $ \{\vec{x}_1,\vec{x}_2, \cdots , \vec{x}_d \} $
$ P_d(e \mid \vec{x})=\int p_d(e \mid \vec{x}, vec{x}_d)p_d(\vec{x}_d ^\' \mid \vec{x}) $