Pattern Recognition Chapter 3: Nearest Neighbour Algorithms
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1 Pattern Recognition Chapter 3: Nearest Neighbour Algorithms Asst. Prof. Dr. Chumphol Bunkhumpornpat Department of Computer Science Faculty of Science Chiang Mai University
2 Learning Objectives What a nearest neighbour (NN) algorithm is The different variants of NN algorithms like The k nearest neighbour (knn) algorithm The modified k nearest neighbour (MkNN) algorithm The r near algorithm The different methods of prototype selection used in classification like The minimal distance classifier (MDC) : Pattern Recognition 2
3 Nearest Neighbour Based Classifiers Simplest Decision Procedures Nearest Neighbour (NN) Rule Similarity between a test pattern and every pattern in a training set Less than twice the probability of error compared to any other decision rule : Pattern Recognition 3
4 Nearest Neighbour Algorithm The nearest neighbour algorithm assigns to a test pattern the class label of its closest neighbour : Pattern Recognition 4
5 Let the training set consist of the following three dimensional patterns: X 1 = (0.8, 0.8, 1), X 2 = (1.0, 1.0, 1), X 3 = (1.2, 0.8, 1) X 4 = (0.8, 1.2, 1), X 5 = (1.2, 1.2, 1), X 6 = (4.0, 3.0, 2) X 7 = (3.8, 2.8, 2), X 8 = (4.2, 2.8, 2), X 9 = (3.8, 3.2, 2) X 10 = (4.2, 3.2, 2), X 11 = (4.4, 2.8, 2), X 12 = (4.4, 3.2, 2) X 13 = (3.2, 0.4, 3), X 14 = (3.2, 0.7, 3), X 15 = (3.8, 0.5, 3) X 16 = (3.5, 1.0, 3), X 17 = (4.0, 1.0, 3), X 18 = (4.0, 0.7, 3) : Pattern Recognition 5
6 Let the training set consist of the following three dimensional patterns: (cont.) Class 1: + Class 2: X Class 3: * P = (3.0, 2.0) d(x, P) = (X[1] P[1]) 2 + (X[2] P[2]) 2 d(x 1, P) = ( ) 2 + ( ) 2 = 2.51 d(x 16, P) = ( ) 2 + ( ) 2 = : Pattern Recognition 6
7 Example Data Set : Pattern Recognition 7
8 Variants of the NN Algorithm k-nearest Neighbour (knn) Algorithm The Modified k Nearest Neighbour (MkNN) Algorithm The r Near Algorithm : Pattern Recognition 8
9 k-nearest Neighbour (knn) Algorithm The majority class of the k nearest neighbours is the class label assigned to the new pattern. The value chosen for k is crucial. With the right value of k, the classification accuracy will be better than that got by using the nearest neighbour algorithm : Pattern Recognition 9
10 Example Data Set (k = 5) : Pattern Recognition 10
11 k-nearest Neighbour (knn) Algorithm (cont.) It reduces the error in classification when training patterns are noisy. The closest pattern of the test pattern may belong to another class, but when a number of neighbours are obtained and majority class label is considered, the pattern is more likely to be classified correctly : Pattern Recognition 11
12 P can be correctly classified using the knn algorithm : Pattern Recognition 12
13 P can be correctly classified using the knn algorithm (cont.) For larger data sets, k can be larger to reduce the error. k can be determined by experimentation, where a number of patterns taken out from the training set (validation set) can be classified using the remaining training patterns for different values of k. It can be chosen as the value which gives the least error in classification : Pattern Recognition 13
14 Example Data Set 2 (k = 5) : Pattern Recognition 14
15 Modified k Nearest Neighbour (MkNN) Algorithm The Distance-Weighted k-nearest Neighbour Algorithm k Nearest Neighbours are weighted according to the distance from the test point : Pattern Recognition 15
16 Modified k Nearest Neighbour (MkNN) Algorithm (cont.) w j = (d k d j ) / (d k d 1 ) if d k d 1 1 if d k = d 1 j = 1,, k w j varies from a maximum of 1 for the nearest neighbour down to a minimum of 0 for the most distant : Pattern Recognition 16
17 Modified k Nearest Neighbour (MkNN) Algorithm (cont.) It assigns the test pattern P to that class for which the weights of the representatives among the k nearest neighbours sums to the greatest value. It employs a weighted majority rule. The outlier patterns have lesser effect on classification : Pattern Recognition 17
18 Example Data Set (k = 5) : Pattern Recognition 18
19 Example Data Set (k = 5) (cont.) d(p, X 16 ) = 1.12 d(p, X 7 ) = 1.13 d(p, X 14 ) = 1.32 d(p, X 6 ) = 1.41 d(p, X 17 ) = : Pattern Recognition 19
20 Example Data Set (k = 5) (cont.) w 16 = 1 w 7 = ( ) / ( ) = 0.97 w 14 = ( ) / ( ) = 0.31 w 6 = 0 w 17 = : Pattern Recognition 20
21 Example Data Set (k = 5) (cont.) Class 1 sums to 0 Class 2 to which X 7 and X 6 belong sums to Class 3 to which X 16, X 14 and X 17 belong sums to The point P belongs to Class 3. knn and MkNN assign the same pattern a same class label : Pattern Recognition 21
22 Example Data Set 2 (k = 5) : Pattern Recognition 22
23 Example Data Set 2 (k = 5) d(p, X 17 ) = 0.83 d(p, X 8 ) = 1.00 d(p, X 11 ) = 1.02 d(p, X 16 ) = 1.06 d(p, X 7 ) = : Pattern Recognition 23
24 Example Data Set 2 (k = 5) (cont.) w 17 = 1 w 8 = ( ) / ( ) = 0.32 w 11 = ( ) / ( ) = 0.24 w 16 = ( ) / ( ) = 0.08 w 7 = : Pattern Recognition 24
25 Example Data Set 2 (k = 5) (cont.) Class 1 sums to 0 Class 2 to which X 8, X 11 and X 7 belong sums to Class 3 to which X 17 and X 16 belong sums to The point P belongs to Class 3. knn and MkNN assign the same pattern different class labels : Pattern Recognition 25
26 r Near Neighbours (rnn) All neighbours within some distance r of the point of interest They ignore very far away nearest neighbour They identify outlier : Pattern Recognition 26
27 r Near Neighbours (rnn) (cont.) Outlier A pattern does not have any similarity with the patterns within the radius chosen. Radius r Crucial : Pattern Recognition 27
28 rnn Algorithm STEP 1: Given the point P, determine the sub-set of data that lies in the ball of radius r centred at P B r (P) = {X i X s.t. P X i r} STEP 2: If B r (P) is empty, then output the majority class of the entire data set. STEP 3: If B r (P) is not empty, output the majority class of the data points in it : Pattern Recognition 28
29 Example Data Set (r = 1.45) : Pattern Recognition 29
30 Prototype Selection Prototype: Sample which would represent or be a typical sample for a large number of samples in the set Centroid Medoid Process of reducing the training set used for classification : Pattern Recognition 30
31 Minimal Distance Classifier (MDC) Each class is represented by the sample mean or centroid of all samples in the class. C i = N j=1 X ij / N X i1, X i2,, X in : N training patterns for the class i If C k is the centroid closest to P, it is assigned the class label k of which C k is the representative pattern : Pattern Recognition 31
32 MDC (cont.) n: The number of train patterns m: The number of test patterns C: The number of classes MDC: O(n + mc) O(n): Time required for computing the centroid O(mC): Time required to search for the nearest centroid of test patterns NN: O(mn) Large-Scale Applications: C << n : Pattern Recognition 32
33 Classification Using MDC : Pattern Recognition 33
34 Centroid Class 1: (1.00, 1.00) Class 2: (4.11, 3.00) Class 3: (3.62, 0.72) : Pattern Recognition 34
35 Classification Using MDC (cont.) From P to the centroid of Class 1, the distance is From P to the centroid of Class 2, the distance is From P to the centroid of Class 3, the distance is : Pattern Recognition 35
36 Reference Murty, M. N., Devi, V. S.: Pattern Recognition: An Algorithmic Approach (Undergraduate Topics in Computer Science). Springer (2012) : Pattern Recognition 36
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