Machine Learning

i=arbitrary

Complex decision boundaries

Machine Learning

Design & Validation of ClassifiersDesign & Validation of Classifiers

Computer Vision

Detectionof

Errors

Sensor

Object

A/D Converter

Pattern of Data

Learning System

Samples Learning System Classifiers

Classification Systems

Data for classification

Classifier Decision Pertaining to class

Design of a classifier

Samples for training

Values ofvariables (xi)

Classes Pertaining to samples

LearningSystem

Classifier type

Classifier forSpecific

application

CaseVariables (Features)

Classes

Estimating the execution of a learning system

What is an error?

Class (+) Class (-)

Classification (+) Correct (+/+) Error (-/+)

Classification (-) Error (+/-) Correct (-/-)

Reason for error (estimate) = number of errors number of cases

Apparent and true error

Classifier

Samplesfor

training

Apparentreason for error

New cases

True error

Error estimationusing samples for training and samples for testing

Cases for training the classifier Cases for testing the classifier

Samples

Sample: Y1i ValueY11 .9846Y12 .-0449Y13 -.7652

Sample: X1i ValueX11 0.7635X12 -1.2667X13 -0.6141X14 1.0913X15 -0.5597

Sample: Y2i ValueY21 .2011Y22 .9438Y23 . 8135

Sample: X2i ValueX21 2.7123X22 1.5558X23 1.8327X24 0.3352X25 0.4325

Example: 1-d

Class 1: n1 = 5

X1 Train

Y1 Train

Class 2: n1 = 5

X2 Train

Y2 Test

Estimation of Parameters

ˆ k 1

X kii1

n i 1X ki ˆ k 2

ˆ 1 .1171ˆ 1

ˆ 21.3737

ˆ 22 1

Classification ML Rule

f X/ wk 1

X1i Value f(X/w1) f(X/w2) ClassX11 .7635 .2707 .3312 2X12 -1.2667 .2060 .0122 1X13 -.6141 .3526 .0553 1X14 1.0913 .1922 .3833 2X15 -.5597 .3617 .0615 1Y11 .9846 .2174 .3699 2Y12 -.0449 .3979 .1459 1Y13 -.7652 .3234 .0405 1

Class 1

f X/ wk 1

X2i Value f(X/w1) f(X/w2) ClassX21 2.7123 .0073 .1629 2X22 1.5558 .0984 .3924 2X23 1.8327 .0596 .3591 2X24 .3352 .3602 .2327 1X25 .4325 .3430 .2562 1Y21 .2011 .3792 .2006 1Y22 .9438 .2273 .3637 2Y23 .8135 .2587 .3410 2

Class 2

A simpler Classification ML Rule

X1i Value ClassX11 .7635 2X12 -1.2667 1X13 -.6141 1X14 1.0913 2X15 -.5597 1Y11 .9846 2Y12 -.0449 1Y13 -.7652 1

Class 1 T ˆ 1 ˆ 2

2.6283

X2i Value ClassX21 2.7123 2X22 1.5558 2X23 1.8327 2X24 .3352 1X25 .4325 1Y21 .2011 1Y22 .9438 2Y23 .8135 2

Class 2 T ˆ 1 ˆ 2

2.6283

Machine Learning

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