1. Fuzzy Logic in Pattern Recognition
Chapter 13

2. Introduction
Pattern recognition techniques can be classified into two broad categories
unsupervised (非監督式) techniques : Clustering.
supervised (監督式) techniques : Classification.
An unsupervised technique use a given set of unclassified data points, where as a supervised technique uses a dataset with known classifications.

3. Unsupervised Clustering
Unsupervised clustering is motivated by the need to find interesting patterns or groupings in a given set of data.
Used to perform the task of “segmenting” the images. (*i.e., partitioning pixels on an image into regions that correspond to different objects in the image)

4. Unsupervised Clustering
Let X be a set of data, and xi be an element of X. A partition P={c1, c2,…, cl}of X is “hard” if and only if
(the partition covers all data points in X)
(all clusters in the partition are mutually exclusive)

5. Unsupervised Clustering
In many real-world clustering problems, some data points partially belong to multiple clusters, rather than to a single cluster exclusive.
Let X be a set of data, and xi be an element of X. A partition P={c1, c2,…, cl} of X is “soft” if and only if
where denotes the degree to which xi belong to cluster cj

6. Unsupervised Clustering
A type of soft clustering of special interest is one that ensures the membership degree of a point X in all clusters adding up to one

7. The (Hard) C-Means Algorithm
The hard c-means algorithm aims to identify compact, well-separated clusters
A compact cluster has a “ball-like” shape. The center of the ball is called the center or the prototype of the cluster
A set of clusters are well separated when any two points in a cluster are closer than the shortest distance between two clusters in different clusters.

8. The (Hard) C-Means Algorithm
Examples
(1) (2)

9. The (Hard) C-Means Algorithm
The goals of hard c-means algorithm is twofold
To find the centers of these clusters
Calculating the current partition based on the current cluster
Modifying the current centers using a gradient descent method to minimize the J function

10. The (Hard) C-Means Algorithm
To determine the clusters (i.e, labels) of each point in the data set
Where vj denotes the center of the cluster cj

11. The (Hard) C-Means Algorithm
The hard C-Means algorithm Step 1: Calculating the current partition based on the current cluster. Step 2: Modifying the current cluster centers using a gradient descent method to minimize the J function.

12. Fuzzy C-means Algorithm
The fuzzy c-means algorithm (FCM) generalizes the hard c-means algorithm to allow a point to partially belong to multiple clusters.

13. Fuzzy C-means Algorithm
Initialize prototype V={v1,v2,…, vc} (每一群中心)
Repeat Vprevious  V
compute membership functions using

14. Fuzzy C-means Algorithm
Update the prototype, vj in V using
until

15. Fuzzy C-means Algorithm
Example

16. Fuzzy C-means Algorithm
The dataset are partitioned into two clusters

17. Classifier Design and Supervised Pattern Recognition
Supervised pattern recognition uses data with known classifications, which are also called labeled data, to determine the classification of new data.
The main benefit of unsupervised pattern recognition techniques is that they do not require training data; however, their computation time is relatively high due to a large number of iterations (疊代) needed before the algorithm converges.

18. Classifier Design and Supervised Pattern Recognition
A supervised pattern recognition technique is relatively fast because it does not need to iterate; however, it cannot be applied to a problem unless training data or relevant knowledge are available.

19. K-Nearest Neighborhood(K-NN)
The main advantage of K-NN is its computational simplicity.
It has two major problems
Each of the neighbors is considered equally important in determining the classification of the input data. A far neighbor is given the same weight as a close neighbor of the input.
The algorithm only assigns a class to the input data, it does not determine the “strength” of membership in the class.

20. Fuzzy K-Nearest Neighborhood
Fuzzy K-NN contains two steps
Find the K-nearest neighbors of the input x
Calculate the class membership of x using