INTRODUCTION TO Machine Learning 3rd Edition ETHEM

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INTRODUCTION TO Machine Learning 3rd Edition ETHEM ALPAYDIN The MIT Press, 2014 alpaydinboun.edu.tr http:www.cmpe.boun.edu.trethemi2ml3e Lecture Slides for CHAPTER 5: Multivariate Methods Multivariate Data 3 Multiple measurements

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INTRODUCTION TO Machine Learning3rd Edition ETHEM ALPAYDIN
© The MIT Press, 2014

alpaydin@boun.edu.tr
http://www.cmpe.boun.edu.tr/~ethem/i2ml3e Lecture Slides for

CHAPTER 5: Multivariate Methods

Multivariate Data 3 Multiple measurements (sensors)
d inputs/features/attributes: d-variate
N instances/observations/examples

Multivariate Parameters 4

Parameter Estimation 5

Estimation of Missing Values 6 What to do if certain instances have missing attributes?
Ignore those instances: not a good idea if the sample is small
Use ‘missing’ as an attribute: may give information
Imputation: Fill in the missing value
Mean imputation: Use the most likely value (e.g., mean)
Imputation by regression: Predict based on other attributes

Multivariate Normal Distribution 7

Multivariate Normal Distribution 8 Mahalanobis distance: (x – μ)T ∑–1 (x – μ)
measures the distance from x to μ in terms of ∑ (normalizes for difference in variances and correlations)
Bivariate: d = 2

Bivariate Normal 9

10

Independent Inputs: Naive Bayes 11 If xi are independent, offdiagonals of ∑ are 0, Mahalanobis distance reduces to weighted (by 1/σi) Euclidean distance:

If variances are also equal, reduces to Euclidean distance

Parametric Classification If p (x | Ci ) ~ N ( μi , ∑i )

Discriminant functions 12

Estimation of Parameters 13

Different Si Quadratic discriminant 14

15 likelihoods posterior for C1 discriminant:
P (C1|x ) = 0.5

Common Covariance Matrix S 16 Shared common sample covariance S

Discriminant reduces to

which is a linear discriminant

Common Covariance Matrix S 17

Diagonal S 18 When xj j = 1,..d, are independent, ∑ is diagonal
p (x|Ci) = ∏j p (xj |Ci) (Naive Bayes’ assumption)

Classify based on weighted Euclidean distance (in sj units) to the nearest mean

Diagonal S 19 variances may be
different

Diagonal S, equal variances 20 Nearest mean classifier: Classify based on Euclidean distance to the nearest mean

Each mean can be considered a prototype or template and this is template matching

Diagonal S, equal variances 21 * ?

Model Selection 22 As we increase complexity (less restricted S), bias decreases and variance increases
Assume simple models (allow some bias) to control variance (regularization)

23

Discrete Features 24 Binary features:
if xj are independent (Naive Bayes’)

the discriminant is linear Estimated parameters

Discrete Features 25 Multinomial (1-of-nj) features: xj Î {v1, v2,..., vnj}

if xj are independent

Multivariate Regression 26 Multivariate linear model

Multivariate polynomial model:
Define new higher-order variables
z1=x1, z2=x2, z3=x12, z4=x22, z5=x1x2
and use the linear model in this new z space
(basis functions, kernel trick: Chapter 13)

INTRODUCTION TO Machine Learning 3rd Edition ETHEM

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