Date post: | 20-Dec-2015 |
Category: |
Documents |
View: | 219 times |
Download: | 0 times |
Greg Grudic Intro AI 1
Introduction to Artificial IntelligenceCSCI 3202Fall 2007
Introduction to ClassificationGreg Grudic
Greg Grudic Intro AI 2
This Class: Classification Models
• Collect Training data• Construct Model: happy = F(feature space)• Make a prediction
HighDimensional
Feature (input)Space
Greg Grudic Intro AI 3
Binary Classification
• A binary classifier is a mapping from a set of d inputs to a single output which can take on one of TWO values (e.g. path/no path)
• In the most general setting
• Specifying the output classes as -1 and +1 is arbitrary!– Often done as a mathematical convenience
{ }inputs:
output:
1, 1
d
y
x Î Â
Î - +
Greg Grudic Intro AI 4
A Binary Classifier
ClassificationModelx ˆ 1, 1y
Given learning data: ( ) ( )1 1, ,..., ,N Ny yx x
A model is constructed:
( )M xNot in learning set!
Greg Grudic Intro AI 5
Classification Learning Data…
Example 1 0.95013 0.58279 1
Example 2 0.23114 0.4235 -1
Example 3 0.8913 0.43291 1
Example 4 0.018504 0.76037 -1
… … … …
1x2x y
Greg Grudic Intro AI 6
The Learning Data
• Matrix Representation of N learning examples of d dimensional inputs
11 1 1
1
d
N Nd N
x x y
x x y
æ ö÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷÷çç ÷è ø
K
M O M M
L
Greg Grudic Intro AI 7
Graphical Representation of 2D Classification Training Data
0 0.2 0.4 0.6 0.8 1 1.20
0.2
0.4
0.6
0.8
1
1.2
x1
x 2
: y=+1: y=-1
Greg Grudic Intro AI 8
Linear Separating Hyper-Planes: Discriminative Classifiers
How many lines can separate these points?
NO!
Greg Grudic Intro AI 9
Linear Separating Hyper-Planes (2 dimensions)
1x
2x
0 1 1 2 2 0x xb b b+ + £
0 1 1 2 2 0x xb b b+ + >
0 1 1 2 2 0x xb b b+ + =
1y =-
1y =+( ) 3
0 1 2ˆ ˆ ˆ, ,b b b Î Â
Greg Grudic Intro AI 10
Linear Separating Hyper-Planes (d dimensions)
1x
2x
01
0d
i ii
xb b=
+ £å
01
0d
i ii
xb b=
+ >å
01
0d
i ii
xb b=
+ =å
1y =-
1y =+( ) 1
0 1ˆ ˆ ˆ, ,..., d
db b b +Î Â
Greg Grudic Intro AI 11
Linear Separating Hyper-Planes
• The Model:
• Where:
• The decision boundary:
( )0 1ˆ ˆ ˆˆ ( ) sgn ,..., dy M x xb b bé ù= = + ×ê úë û
[ ]1 if 0
sgn1 otherwise
AA
ì >ïï=íï -ïî
( )0 1 01
ˆ ˆ ˆ,..., 0d
d i ii
xxb b b b b=
+ × = + =å
Greg Grudic Intro AI 12
Linear Separating Hyper-Planes
• The model parameters are:
• The hat on the betas means that they are estimated from the data
• Many different learning algorithms have been proposed for determining
( ) 10 1
ˆ ˆ ˆ, ,..., ddb b b +Î Â
( )0 1ˆ ˆ ˆ, ,..., db b b
Is this Data Linearly Separable?
Greg Grudic Intro AI 13
NO!
Is this Data Linearly Separable?
Greg Grudic Intro AI 14
YES!
Is this Data Linearly Separable?
Greg Grudic Intro AI 15
NO!
Is this Data Linearly Separable?
Greg Grudic Intro AI 16
YES!