Linear Methods for Classification

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Linear Methods for

Classification

Lecture Notes for CMPUT 466/551

Nilanjan Ray

1

Linear Classification

• What is meant by linear classification?

– The decision boundaries in the in the feature

(input) space is linear

• Should the regions be contiguous?

R

1 R

2

X

2 R

3

R

4

X

1

Piecewise linear decision boundaries in 2D input space 2

Linear Classification…

• There is a discriminant function

 k

( x ) for each class k

• Classification rule:

R k

{ x : k

 arg max

 j

( x )} j

• In higher dimensional space the decision boundaries are piecewise hyperplanar

• Remember that 0-1 loss function led to the classification rule:

• So,

P ( G

 k | X )

R k

{ x : k

 arg max can serve as

 j k

( x )

P ( G

 j | X

 x )}

3

Linear Classification…

• All we require here is the class boundaries

{ x :

 k

( x ) =

 j

( x )} be linear for every ( k , j ) pair

• One can achieve this if  k

( x ) themselves are linear or any monotone transform of

 k

( x ) is linear

– An example:

P ( G

1 | X

 x )

 exp(

1

 exp(

0

0

 

T

 x )

T x )

So that

P ( G log[

2 | X

P ( G

P ( G

1 |

2 |

X

X x )

1

 exp(

1

0

 x ) x )

]

 

0

 

T x

T x )

Linear 4

Linear Classification as a Linear

Regression

2D Input space: X = ( X

1

, X

2

)

Number of classes/categories K =3, So output Y = ( Y

1

, Y

2

, Y

3

)

Training sample, size N =5,

X

1

 1

1

1

1 x

11 x

21 x

31 x

41 x

51 x

12 x

22 x

32 x

42 x

52

, Y

 y y y

11

21

31 y

41 y

51 y

12 y

22 y

32 y

42 y

52 y y

23 y

33 y

13

43 y

53

Each row has exactly one 1 indicating the category/class

Indicator Matrix

Regression output: Y

ˆ

(( x

1

, x

2

))

( 1 x

1 x

2

)( X

T

X )

1

X

T

Y

( x

T

1 x

T

2 x

T

3

)

Or, Y

ˆ

1

(( x

1

Y

ˆ

2

(( x

1

Y

ˆ

3

(( x

1 x x

2

2

))

)) x

2

))

( 1

(

( 1

1 x

1 x x

1

1 x

2

)

1 x

2

)

2 x

2

)

3

Classification rule:

ˆ

(( x

1 x

2

))

 arg max k

Y

ˆ k

(( x

1 x

2

))

5

The Masking

Linear regression of the indicator matrix can lead to masking

2D input space and three classes Masking

Y

ˆ

2

( 1 x

1 x

2

)

2

Y

ˆ

3

( 1 x

1 x

2

)

3

Y

ˆ

1

( 1 x

1 x

2

)

1

Viewing direction

LDA can avoid this masking

6

Linear Discriminant Analysis

Essentially minimum error Bayes’ classifier

Assumes that the conditional class densities are ( multivariate ) Gaussian

Assumes equal covariance for every class

Posterior probability Pr( G

 k | X

 x )

 l

K 

1 f k

( x )

 k f l

( x )

 l

 k is the prior probability for class k f k

( x ) is class conditional density or likelihood density f k

( x )

1

( 2

) p / 2

| Σ |

1 / 2 exp(

1

2

( x

  k

)

T Σ 

1

( x

  k

))

Application of

Bayes rule

7

LDA… log

Pr( G

Pr( G

 k l

|

|

X

X

 x x )

)

 log

 k l

 log f k f l

(log

 k

 x

T Σ 

1

 k

1

2

 k

T Σ 

1

 k

)

(log

 l

 x

T Σ 

1

 l

1

2

 l

T Σ 

1

 l

)

 k

( x )

 l

( x )

Classification rule:

ˆ

( x )

 arg max k

 k

( x ) is equivalent to:

ˆ

( x )

 arg max k

Pr( G

 k | X

 x )

The good old Bayes classifier!

8

LDA…

When are we going to use the training data?

Total N input-output pairs

N k number of pairs in class

Total number of classes: K k

Training data utilized to estimate

Prior probabilities:

Means:

Covariance matrix:

( g i

, x i

), i

1 : N

 ˆ  k

N k

/ N

 ˆ k

  g i

 k x i

/ N k

   k

1 g i

( x i

  ˆ k

)( x i

  ˆ k

)

T

/( N

K )

9

LDA: Example

LDA was able to avoid masking here

10

Quadratic Discriminant Analysis

• Relaxes the same covariance assumption– class conditional probability densities (still multivariate Gaussians) are allowed to have different covariant matrices

• The class decision boundaries are not linear rather quadratic log

Pr( G

Pr( G

 k l |

| X

X

 x x )

)

 log

 k l

 log f k f l

(log

 k

1

2

( x

  k

)

T Σ k

1

( x

  k

)

1

2 log |

Σ k

|)

(log

 l

1

2

( x

  l

)

T Σ l

1

( x

  l

)

1

2 log |

Σ l

|)

 k

( x )

 l

( x )

11

QDA and Masking

Better than Linear Regression in terms of handling masking:

Usually computationally more expensive than LDA

12

Fisher’s Linear Discriminant

[DHS]

From training set we want to find out a direction where the separation between the class means is high and overlap between the classes is small

13

Fisher’s LD…

Projection of a vector x on a unit vector w : w T x

Geometric interpretation: x w w

T x

From training set we want to find out a direction w where the separation between the projections of class means is high and the projections of the class overlap is small

14

Fisher’s LD…

Class means: m

1

1

N

1 x i

R

1 x i

,

Projected class means: m

2

1

N

2 x i

R

2 x i

~

1

1

N

1 x i

R

1 w T x i

 w T m

1

,

Difference between projected class means:

~

2

1

N

2 x i

R

2 w T x i

 w T m

2

2

1

 w

T

( m

2

 m

1

)

Scatter of projected data (this will indicate overlap between the classes):

~

1

2

~

2

2

 y i

: x i

R

1

( y i

 y i

: x i

R

2

( y i

~

1

) 2

~

2

) 2

 x i

R

1

( w T x i x i

R

2

( w T x i

 w T m

1

) 2

 w T m

2

) 2

 w T

 x i

R

1

( x i

 w T

 x i

R

2

( x i

 m

1

)( x i

 m

2

)( x i

 m

1

) T

 m

2

) T

 w

 w

 w T S

1 w

 w T S

2 w

15

Fisher’s LD…

Ratio of difference of projected means over total scatter: r ( w )

(

~

2

~

1

2

~

1

~

2

2

)

2

 w

T

S

B w w

T

S w w

Rayleigh quotient where S w

S

1

S

2

S

B

( m

2

 m

1

)( m

2

 m

1

)

T

We want to maximize r ( w ). The solution is w

S w

1

( m

2

 m

1

)

16

Fisher’s LD: Classifier

So far so good. However, how do we get the classifier?

All we know at this point is that the direction w

S w

1

( m

2

 m

1

) separates the projected data very well

Since we know that the projected class means are well separated, we can choose average of the two projected means as a threshold for classification

Classification rule: x in R

2 if y ( x )>0, else x in R

1

, where y ( x )

 w

T x

1

2

(

1

2

)

 w

T x

1

2 w

T

( m

1

 m

2

)

S w

1

( m

2

 m

1

)( x

1

2

( m

1

 m

2

))

17

Fisher’s LD and LDA

They become same when

(1) Prior probabilities are same

(2) Common covariance matrix for the class conditional densities

(3) Both class conditional densities are multivariate Gaussian

Ex.

Show that Fisher’s LD classifier and LDA produce the same rule of classification given the above assumptions

Note: (1) Fisher’s LD does not assume Gaussian densities

(2) Fisher’s LD can be used in dimension reduction for a multiple class scenario

18

Logistic Regression

• The output of regression is the posterior probability i .

e ., Pr(output | input)

• Always ensures that the sum of output variables is 1 and each output is non-negative

• A linear classification method

• We need to know about two concepts to understand logistic regression

– Newton-Raphson method

– Maximum likelihood estimation

19

Newton-Raphson Method

A technique for solving non-linear equation f ( x )=0

Taylor series: f ( x n

1

)

 f ( x n

)

( x n

1

 x n

) f

( x n

)

After rearrangement: x n

1

 x n

 f ( x n f

1

)

(

 x n

) f ( x n

)

If x n+1 is a root or very close to the root, then: f ( x n

1

)

0

So: x n

1

 x n

 f f (

( x n x n

)

)

Rule for iteration

Need an initial guess x

0

20

Newton-Raphson in Multi-dimensions

We want to solve the equations:

Taylor series: f f

1

( x

1

, x

2

,  , x

N

)

0

2

( x

1

, x

2

,  , x

N

)

0

 f

N

( x

1

, x

2

,  , x

N

)

0 f j

( x

  x )

 f j

( x )

 k

N 

1

 f j

 x k

 x k

, j

1 ,..., N

After some rearrangement etc.

the rule for iteration:

(Need an initial guess)

 x

1 x

2 n n

1

1 x

N n

1

 x

1 x n

1 n

1

2

 x

N n

1

  x

1

 f

N

 x

1

 f

1

 f x

1

2

 f

1

 x f

2

2

 x

 f

N

2

 x

2

 f

1

 x f

N

2

 x

 f

N

N

 x

N

1

 f f f

1

(

2

N

(

( x x

1 n

1 x

1 n n ,

,

, x x n

2 n

2

 x

2 n

,

,

,

,  ,

, x x n

N n

N x

N n

)

)

)

Jacobian matrix

21

Newton-Raphson : Example

Solve: f

1

( x

1

, x

2

)

 x

1

2 f

2

( x

1

, x

2

)

 sin(

 cos( x

2

)

0 x

1

)

 x

1

2  x

2

3 

0

 x

1 n

1 x

2 n

1

 x

1 x

2 n n

 cos(

2 x

1 n x

1 n

)

2 x

1 n sin(

3 ( x

2 n x

2 n

)

2

)

1 

 sin(

( x

1 n x

1 n

)

) 2

(

 x cos( x

2 n

1 n

)

2 

(

) x

2 n

)

3

Iteration rule need initial guess

22

Maximum Likelihood Parameter

Estimation

Let’s start with an example. We want to find out the unknown parameters mean and standard deviation of a Gaussian pdf, given N independent samples from it.

p ( x ;

,

)

1

2

  exp(

( x

2

)

2

2

)

Samples: x

1

,…., x

N

Form the likelihood function : L (

,

)

 i

N 

1

1

2

  exp(

( x i

2

2

)

2

)

Estimate the parameters that maximize the likelihood function

( ,

 ˆ

)

 arg

 max

,

L (

,

)

Let’s find out

(

 ˆ

,

 ˆ

) 23

Logistic Regression Model

The method directly models the posterior probabilities as the output of regression

Pr( G

 k | X

 x )

1

 exp(

 k 0 l

K 

1

1 exp(

 l

0

 k

T

 x )

 l

T x )

,

Pr( G

K | X

 x )

1

1 l

1 K 

1 exp(

 l 0

  l

T x ) k

1 ,  , K

1 x is p -dimensional input vector

 k is a p -dimensional vector for each k

Total number of parameters is ( K -1)( p +1)

Note that the class boundaries are linear

How can we show this linear nature?

What is the discriminant function for every class in this model?

24

Logistic Regression Computation

Let’s fit the logistic regression model for K =2, i.e., number of classes is 2

Training set: ( x i

, g i

), i=1,…, N

Log-likelihood: l (

)

 i

N 

1

{log Pr( G

 y i

| X

 x i

)}

 i

N 

1 y i log(Pr( G

1 | X

 x i

))

( 1

 y i

) log(Pr( G

0 | X

 x i

))

 i

N 

1

( y i

T x i

( 1

 y i

) log

1

1 exp(

T x i

)

)

 i

N 

1

( y i

T x i

( 1

 y i

) log( 1

 exp(

T x i

))) x i are ( p+ 1 ) -dimensional input vector with leading entry 1

 is a ( p+ 1 ) -dimensional vector y i

= 1 if g i

=1; y i

= 0 if g i

=2

We want to maximize the log-likelihood in order to estimate

 25

Newton-Raphson for LR

 l (

 

)

 i

N 

1

( y i

 exp(

1

 exp(

T

 x )

T x )

) x i

0

( p +1) Non-linear equations to solve for ( p +1) unknowns

Solve by Newton-Raphson method:

   

[ Jacobian(

 l (

)

 

)]

1

 l (

)

 

, where, Jacobian(

 l (

 

)

)

i

N 

1 x i x i

T exp(

(

1

 exp(

T

 x i

T

) x i

)

)(

1

1 exp(

T x i

)

)

26

NewtonRaphson for LR…

 l (

 

)

 i

N 

1

( y i

 exp(

1

T exp(

 x )

T x )

) x i

X

T

( y

 p )

Jacobian (

 l (

 

)

)

 

X

T

WX

So, NR rule becomes:

   

( X

T

WX )

1

X

T

( y

 p ),

X

 x

1

T x x

T

2

T

N

N

 by

( p

1 )

, y

 y y

 y

N

1

2

N

 by

1

, p

 exp( exp(

 exp(

T

T

T x x x

1

N

2

)

)

) /( 1

/( 1

/( 1

 exp(

 exp(

 exp(

T

T

T x

1 x

2 x

N

))

))

))

N

 by

1

,

W is a N -byN diagonal matrix with i th diagonal entry: exp(

(

(1

 exp(

T

 x i

T

) x i

) )

)( 1

(1

1 exp(

T x i

) )

) 27

NewtonRaphson for LR…

• Newton-Raphson

–  new   old 

( X

T

WX )

1

X

T

( y

 p )

( X

T

WX )

1

X

T

W ( X

 old 

W

1

( y

 p ))

( X

T

WX )

1

X

T

Wz

– Adjusted response z

X

 old 

W

1

( y

 p )

– Iteratively reweighted least squares (IRLS)

 new  arg min

( z

X

T

)

T

W ( z

X

T

)

 arg min

( y

 p )

T

W

1

( y

 p )

28

Example: South African Heart

Disease

29

Example: South African Heart

Disease…

After data fitting in the logistic regression model:

Pr( MI

 yes | x )

1 exp(

4 exp(

.

130

4 .

130

0

.

006 x

0 .

006 sbp x

 sbp

0 .

08 x tobaco

0 .

08 x tobaco

0

.

185 x

0 .

185 ldl x

 ldl

0 .

939 x famhist

0 .

939 x famhist

0

.

035 x

0 .

035 obesity x

 obesity

0 .

001 x alcohol

0 .

001 x

 alcohol

0

.

043

0 .

x age

043 x

) age

)

(Intercept) sbp tobacco ldl famhist obesity alcohol age

Coefficient Std. Error Z Score

-4.130

0.964

-4.285

0.006

0.006

1.023

0.080

0.185

0.939

-0.035

0.001

0.043

0.026

0.057

0.225

0.029

0.004

0.010

3.034

3.219

4.178

-1.187

0.136

4.184

30

Example: South African Heart

Disease…

After ignoring negligible coefficients:

Pr( MI

 yes | x )

1 exp(

4 exp(

.

204

4 .

204

0 .

081 x tobaco

0 .

081 x

 tobaco

0

.

0

168 x

.

168 ldl x

 ldl

0

.

924

0 .

x famhist

924 x famhist

0

.

044

0 .

x age

044 x

) age

)

What happened to systolic blood pressure? Obesity?

31

Multi-Class Logistic Regression

NR update:

~

 

~

(

~

X

T

~

W

~

X )

1

~

X

T

(

)

~

( K

( K

10

11

1

2 p

20 p

1 ) 0

1 ) p

( K

1 )( p

1 )

 by

1

,

~

X

X

X

X

X

N ( K

1 )

 by

( K

1 )( p

1 )

, where X

1

 1

 1 x x

T

1

T

2 x

T

N

N

 by

( p

1 )

32

MultiClass LR…

is a N ( K -1) dimension vector: 

 y y

1

2 y

K

1

, where y k

(

(

( g

1 g g

N

2

 k k k

)

)

)

, 1

 k

K

1 .

( z ) is a indicator function: 

( z )

1 ,

0 , if z

0 otherwise .

is a N ( K -1) dimension vector:

 p p

1

2 p

K

1

, where p k

 exp(

 exp(

 exp(

 k k k

0

0

0

 

 k k k x

1 x

2 x

N

) /( 1

) /( 1

) /( 1

K 

1 l

K

1

1 l

1 exp(

 l 0 exp(

 l 0 l

K 

1

1 exp(

 l 0

 l l l x x

1

2 x

N

) )

) )

) )

, 1

 k

K

1 .

33

MCLR…

W

W

W

 W

( K

11

21

1 ) 1

W

12

W

22

W

( K

1 ) 2

W

1 (

W

2 (

K

1 )

K

1 )

W

( K

1 )( K

1 )

N ( K

1 )

 by

N ( K

1 )

, where W km

, 1

 k , m

K

1 , is an N

 by

N diagonal matrix, if k

 m , then the i th diagonal entry is (

(1

 exp(

 k 0

 l

K  1

1 exp(

 l 0

 k

T

 x i

)

 l

T x i

) )

)( 1

(1

 exp(

 k 0

 l

K  1

1 exp(

 l 0

 k

T

 x i

)

 l

T x i

) )

), if k

 m , then the i th diagonal entry is

(

(1

 l exp(

 k 0

K  1

1

 exp(

 l 0

 k

T

 x i

)

 l

T x i

) )

)(

(1

 exp(

 m 0 l

K  1

1

 exp(

 l 0

 m

T x i

)

  l

T x i

) )

).

34

LDA vs. Logistic Regression

• LDA (Generative model)

– Assumes Gaussian class-conditional densities and a common covariance

– Model parameters are estimated by maximizing the full log likelihood, parameters for each class are estimated independently of other classes, Kp+p ( p+1 ) /2+ ( K-1 ) parameters

– Makes use of marginal density information Pr( X )

– Easier to train, low variance, more efficient if model is correct

– Higher asymptotic error, but converges faster

• Logistic Regression (Discriminative model)

– Assumes class-conditional densities are members of the (same) exponential family distribution

– Model parameters are estimated by maximizing the conditional log likelihood, simultaneous consideration of all other classes, ( K-1 )( p+1 ) parameters

– Ignores marginal density information Pr( X )

– Harder to train, robust to uncertainty about the data generation process

– Lower asymptotic error, but converges more slowly

Generative vs. Discriminative Learning

Example

Generative

Linear Discriminant

Analysis

Discriminative

Logistic Regression

Objective Functions Full log likelihood:

 i log p

( x i

, y i

)

Model Assumptions Class densities:

( x | y e.g. Gaussian in LDA

Parameter Estimation

Conditional log likelihood

 i log p

( y i

| x i

)

Discriminant functions k )

 k

( x )

“Easy” – One single sweep “Hard” – iterative optimization

Advantages

Disadvantages

More efficient if model correct, borrows strength from p(x)

More flexible, robust because fewer assumptions

Bias if model is incorrect May also be biased. Ignores information in p(x)

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