111

Predictive Learning from Data

Electrical and Computer Engineering

LECTURE SET 7

Methods for Regression

2

OUTLINE of Set 7• Objectives

- introduce taxonomy of methods for regression;- describe several representative nonlinear methods;- empirical comparisons illustrating advantages and limitations of these methods

• Methods taxonomy• Linear methods• Adaptive dictionary methods• Kernel methods and local risk minimization• Empirical comparisons• Combining methods• Summary and discussion

3

Motivation and issues• Importance of regression for implementation of

- classification- density estimation

• Estimation of a real-valued function when data (x,y) is generated as

• Major issues for regression- parameterization (representation) of f(x,w)- optimization formulation (~ empirical loss)- complexity control (model selection)

• These issues are inter-related

noisegy )(x

4

Loss function and noise model• Fundamental problem: how to distinguish

between true signal and noise?

• Classical statistical view- noise density p(noise) is known statistically optimal loss function in the maximum likelihood sense is

for Gaussian noise use squared loss (MSE) as empirical loss function

noisegy )(x

)),(()),(,( wfylogpwxfyL x

5

Loss functions for linear regression• Consider linear regression only

• Several unimodal noise models:- Gaussian, Laplacian, unimodal

• Statistical view:- Optimal loss for known noise density- asymptotic setting- robust strategies when noise model unknown

• Practical situations- noise model unknown- finite (sparse) sample setting

xwx 0),( wwf

6

(a)Linear loss for Laplacian noise (b)Squared loss for Gaussian noise

-3 -2 -1 0 1 2 30

0.5

1

1.5

2

2.5

3

-3 -2 -1 0 1 2 30

0.5

1

1.5

2

2.5

3

7

-insensitive loss (SVM) has common-sense interpretation.Optimal epsilon depends on noise level and sample size

-3 -2 -eps 0 eps 2 3 0

0.5

1

1.5

2

2.5

3

8

Comparison for high-dimensional data:

Gaussian noise Laplacian noise

30,1 n20]1,0[x20214 )( xxxg x

OLS LM SVM.0

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2

3

4

5

6

7

8

9

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OLS LM SVM.0

1

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8

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OLS LM SVM0

1

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6

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8

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OLS LM SVM0

1

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Methods’ Taxonomy• Recall implementation of SRM:

- fix complexity (VC-dimension)

- minimize empirical risk (squared-loss)

• Two interrelated issues:

- parameterization (of possible models)

- optimization method

• Taxonomy will be based on

parameterization: dictionary vs kernel

flexibility: non-adaptive vs adaptive

10

• Dictionary representation

Two possibilities• Linear (non-adaptive) methods

~ predetermined (fixed) basis functions

only parameters have to be estimated

via standard optimization methods (linear least squares)

Examples: linear regression, polynomial regression

linear classifiers, quadratic classifiers• Nonlinear (adaptive) methods

~ basis functions depend on the training data

Possibilities : nonlinear b.f. (in parameters ), i.e. MLP

feature selection (i.e. wavelet denoising)

xig

g x,vi

wi

f m x , w,V wi g x,vi i 0

m

iv

11

Example Nonlinear Parameterizations• Basis functions of the form

i.e. sigmoid aka logistic function

- commonly used in artificial neural networks- combination of sigmoids ~ universal approximator

)()( iii bgtg xv

tt

exp1

1s

12

Neural Network Representation• MLP or RBF networks

- dimensionality reduction- universal approximation property – see example at http://www.mathworks.com/products/demos/nnettlbx/radial/index.html

W is m 1

1 2 m

V is d m

ˆ y w jz jj 1

m

zj g x ,v j

x1 x2 xd

z1 z2 zm

f m x , w,V wi g x,vi i 0

m

13

Kernel Methods• Model estimated as

where symmetric kernel function is

- non-negative

- radially symmetric

- monotonically decreasing with• Duality between dictionary and kernel representation:

model ~ weighted combination of basis functions

model ~ weighted combination of output values• Selection of kernel functions

non-adaptive ~ depends only on x-values

adaptive ~ depends on y-values of training data

Note: kernel methods may require local complexity control

f x Ki x ,x i i 1

n

yi

K x,x' 0K x,x' K x x'

t x x' limt

K t 0

Ki x ,xi

14

OUTLINE• Objectives• Methods taxonomy• Linear methods

Estimation of linear modelsEquivalent RepresentationsNon-adaptive methodsApplication Example

• Adaptive dictionary methods• Kernel methods and local risk minimization• Empirical comparisons• Combining methods• Summary and discussion

15

Estimation of Linear ModelsDictionary representation

• Parameters w estimated via least-squares• Denote training data as matrix

and vector of response values • OLS solution ~ solving matrix equation

where

m

iiim w

0

g,f xwx

y y1 , . . . ,yn X x1 , .. . , xn

Zw y

Z

g1 x1 . .. gm x1 .. . . . .. . . .. . . . .. . . . .. .

g1 xn . . .gm xn

g1 X g2 X . . . gm X

Remp w 1

nZw y

2

16

Estimation of Linear Models (cont’d)• Solution exists if the columns of Z are linearly

independent (m < n)

• Solving the normal equation

yields OLS solution

• Similar math holds for penalized OLS where

OLS solution

Zw y

ZTZw ZTy

w * ZTZ 1ZTy

Rpen w 1

nZw y

2 wTw w * ZTZ 1

ZTy

17

For dictionary representationOLS solution is

nXn matrix S ~projection matrix

Matrix S ~ ‘equivalent kernel’ of an OLS model w*

f m x , w,V wi g x,vi i 0

m

Equivalent Representation

Column Space of Z

y

y Zw*

ˆ y Zw* Sy

ˆ y Zw* Sy

S Z ZTZ 1ZT

S x, xi g x ZTZ 1gT x i

18

• Equivalent kernel

may not be local• Equivalent ‘kernels’ of a 3-rd degree polynomial

Equivalent Representation (cont’d)

ˆ y Zw* SyS x, xi g x ZTZ 1

gT x i

-0.1

0

0.1

0.2

0.3

0.4

0 0.2 0.4 0.6 0.8 1

19

• Eigenfunction decomposition of a kernel

• The eigenvalues tend to fall off rapidly with I

4 BF’s for kernel

Equivalent BFs for Symmetric Kernel

11

gggi

iii

iii weK xxxxx,

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 0.2 0.4 0.6 0.8 1

e1 1.0

e2 0.45

e3 0 .10

e4 0.02

2

2

)55.0(2

texptg

20

• Equivalence of representations is due to duality of OLS solution

• Equivalent ‘kernels’ are just math artifacts (may be

non-local). Notational distinction: K vs S

• Practical use of matrix S for:

- analytic form of LOO cross-validation

- estimating model complexity for penalized linear estimators (ridge regression)

Equivalent Representation: summary

ˆ y Zw* Sy

21

• Linear estimator is specified via matrix S. Its complexity ~ the number of parameters m of an equivalent linear estimator

ave variance of training data

• Consider an equivalent linear estimator with matrix where is symmetric of rank m :

so the average variance is

effective DoF of estimator with matrix S is

Estimating Complexity

222

22ˆˆˆvar

Tiiii

iiiii

EEE

EEyEyEy

sssyys

ysys

var ˆ y 2

ntrace SST

˜ S

trace ˜ S S T trace ˜ S rank ˜ S m

var ˆ y 2m

n TSStraceDoF

˜ S

22

Non-adaptive methods• Dictionary representation

basis functions depend only on x-values

• Representative methods include:

- local polynomials (splines) from statisticswhere parameters are knot locations

- RBF networks from neural networkswhere parameters are RBF center and width

Only non-adaptive implementation of RBF will be considered

f m x , w,V wi g x,vi i 0

m

g x,vi

iv

iv

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Local polynomials and splines• Problem setting: data interpolation(univariate)

problem with polynomials local low-order polynomials

knot location strategies: subset of training samples, or uniformly spaced in x-domain.

-0.5

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

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RBF Networks for Regression• RBF networks

typically local BFs

• Training ~ estimating-parameters of BF’s-linear weights W

- non-adaptive implementation (TBD)- adaptive implementation

W is m 1

1 2 m

V is d m

ˆ y w jz jj 1

m

zj g x ,v j

x1 x2 xd

z1 z2 zm

f m x , w w j gx v j

j

j 1

m

w0

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Non-adaptive RBF training algorithm1. Choose the number of basis functions

(centers) m.2. Estimate centers using x-values of training data

via unsupervised training (SOM, GLA, clustering etc.)3. Determine width parameters using heuristic:

For a given center (a) find the distance to the closest center:

for all

(b) set the width parameter where parameter controls degree of overlap between adjacent basis functions. Typically

4. Estimate weights w via linear least squares (minimization of the empirical risk).

v j

v j

j

r j mink

vk v jk j

j rj1 3

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Application Example: Predicting NAV of Domestic Mutual Funds

• Motivation

• Background on mutual funds

• Problem specification + experimental setup

• Modeling results

• Discussion

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Background: pricing mutual funds• Mutual funds trivia• Mutual fund pricing:

- priced once a day (after market close) NAV unknown when order is placed

• How to estimate NAV accurately? Approach 1: Estimate holdings of a fund (~200-400 stocks), then find NAVApproach 2: Estimate NAV via correlations btwn NAV and major market indices (learning)

28

Problem specs and experimental setup

• Domestic fund: Fidelity OTC (FOCPX)• Possible Inputs:

SP500, DJIA, NASDAQ, ENERGY SPDR• Data Encoding:

Output ~ % daily price change in NAV

Inputs ~ % daily price changes of market indices

• Modeling period: 2003.

• Issues: modeling method? Selection of input variables? Experimental setup?

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Experimental Design and Modeling Setup

Mutual FundsMutual Funds Input VariablesInput Variables

YY X1X1 X2X2 X3X3

FOCPXFOCPX ^IXIC^IXIC -- --

FOCPXFOCPX ^GSPC^GSPC ^IXIC^IXIC --

FOCPXFOCPX ^GSPC^GSPC ^IXIC^IXIC XLEXLE

• All variables represent % daily price changes.• Modeling method: linear regression• Data obtained from Yahoo Finance.• Time period for modeling 2003.

Possible variable selection:

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Specification of Training and Test Data

Year 2003

1, 2 3, 4 5, 6 7, 8 9, 10 11, 12

Training Test

Training Test

Training Test

Training Test

Training Test

Two-Month Training/ Test Set-up Total 6 regression models for 2003

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Results for Fidelity OTC Fund (GSPC+IXIC)

Coefficients w0 w1 (^GSPC) W2(^IXIC)

Average -0.027 0.173 0.771

Standard Deviation (SD) 0.043 0.150 0.165

Average model: Y =-0.027+0.173^GSPC+0.771^IXIC^IXIC is the main factor affecting FOCPX’s daily price change Prediction error: MSE (GSPC+IXIC) = 5.95%

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Results for Fidelity OTC Fund (GSPC+IXIC)

Daily closing prices for 2003: NAV vs synthetic model

80

90

100

110

120

130

140

1-Jan-03 20-Feb-03 11-Apr-03 31-May-03 20-Jul-03 8-Sep-03 28-Oct-03 17-Dec-03

Date

Daily A

cco

un

t V

alu

e

FOCPX

Model(GSPC+IXIC)

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Average Model: Y=-0.029+0.147^GSPC+0.784^IXIC+0.029XLE ^IXIC is the main factor affecting FOCPX daily price change Prediction error: MSE (GSPC+IXIC+XLE) = 6.14%

Coefficients w0 w1 (^GSPC) W2(^IXIC) W3(XLE)

Average -0.029 0.147 0.784 0.029

Standard Deviation (SD) 0.044 0.215 0.191 0.061

Results for Fidelity OTC Fund (GSPC+IXIC+XLE)

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Results for Fidelity OTC Fund (GSPC+IXIC+XLE)

Daily closing prices for 2003: NAV vs synthetic model

80

90

100

110

120

130

140

1-Jan-03 20-Feb-03 11-Apr-03 31-May-03 20-Jul-03 8-Sep-03 28-Oct-03 17-Dec-03

Date

Da

ily

Ac

co

un

t V

alu

e

FOCPX

Model(GSPC+IXIC+XLE)

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Effect of Variable Selection

Different linear regression models for FOCPX:• Y =-0.035+0.897^IXIC

• Y =-0.027+0.173^GSPC+0.771^IXIC• Y=-0.029+0.147^GSPC+0.784^IXIC+0.029XLE• Y=-0.026+0.226^GSPC+0.764^IXIC+0.032XLE-0.06^DJI

Have different prediction error (MSE):• MSE (IXIC) = 6.44%• MSE (GSPC + IXIC) = 5.95%• MSE (GSPC + IXIC + XLE) = 6.14%• MSE (GSPC + IXIC + XLE + DJIA) = 6.43%

(1) Variable Selection is a form of complexity control

(2) Good selection can be performed by domain experts

36

Discussion• Many funds simply mimic major indices statistical NAV models can be used for

ranking/evaluating mutual funds

• Statistical models can be used for

- hedging risk and

- to overcome restrictions on trading (market timing) of domestic funds

• Since 70% of the funds under-perform their benchmark indices, better use index funds

37

OUTLINE• Objectives• Methods taxonomy• Linear methods• Adaptive dictionary methods

- additive modeling and projection pursuit- MLP networks- CART and MARS

• Kernel methods and local risk minimization• Empirical comparisons• Combining methods• Summary and discussion

38

Additive Modeling & Projection Pursuit

• Additive models have parameterization for regression

where is an adaptive basis function

• Backfitting is a greedy optimization approach for estimating basis functions sequentially:

- basis function is estimated by holding all other basis functions fixed

f x,V g j x,v j j 1

m

w0

g j x,v j

kkg vx,j k

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• By fixing all basis functions the empirical risk (MSE) can be decomposed as

Each basis function is estimated via an iterative backfitting algorithm (until some stopping criterion is met)

Note: can be interpreted as the response variable for the adaptive method

j k

n

ikiki

n

ikik

kjjiji

n

iiiemp

rn

wyn

yn

R

1

2

1

2

0

1

2

g1

gg1

f1

v,x

v,xv,x

V,xV

kkg vx,

ir kk vx,g

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• Consider regression estimation of a function of two variables of the form

from training data

For example

Backfitting method: (1) estimate for fixed

(2) estimate for fixed

iterate above two steps• Estimation via minimization of empirical risk

ni ,...,2,1

n

iii

n

iiii

n

iiiiemp

xgrn

xgxgyn

stepfirst

xgxgyn

xgxgR

1

211

1

21122

1

222112211

)(1

)()(1

)_(

)()(1

)(),(

noisexgxgy 2211

Backfitting Algorithm

)2sin(),( 22121 xxxxt

),,( 21 iii yxx 21,0x

11g x 2g 22g x 1g

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• Estimation of via minimization of MSE

• This is a univariate regression problem of estimating from n data points

where • Can be estimated by smoothing (kNN regression)• Estimation of (second_step) proceeds in a

similar manner, via minimization of

where

minxgrn

xgRn

iiiemp

1

21111 )(

1)(

Backfitting Algorithm(cont’d)

)( 22 iii xgyr 11g x

22g x

)(g 11 x

),( 1 ii rx

n

iiiemp xgr

nxgR

1

22222 )(

1)( )( 11 iii xgyr

42

Projection Pursuit regression• Projection Pursuit is an additive model:

where basis functions are univariate functions (of projections)

• Backfitting algorithm is used to estimate iteratively(a) basis functions (parameters ) via scatterplot smoothing(b) projection parameters (via gradient descent)

f x,V, W g j w j x ,v j j 1

m

w0

g j z,v j

w j

v j

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EXAMPLE: estimation of a two-dimensional fct via projection pursuit

(a) Projections are found that minimize unexplained variance. Smoothing is performed to create adaptive basis functions.

(b) The final model is a sum of two univariate adaptive basis functions.

-1.5

-1

-0.5

0

0.5

1

-2 -1 0 1 2

g1 z1 r

z1 w1x

g2 z2 r

z2 w2 x

-1

0

1

2

3

4

5

6

-2 -1 0 1 2

x1 x2

z1

z2

y

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Multilayer Perceptrons (MLP)• Recall MLP networks

for regressionwhere

or

• Parameters (weights) estimated via backpropagation

W is m 1

1 2 m

V is d m

ˆ y w jz jj 1

m

zj g x ,v j

x1 x2 xd

z1 z2 zmg x ,vi s vi 0 xkvikk1

d

s x v i

s t 1

1 exp t

s t tanh t exp t exp t exp t exp t

45

Gradient Descent Learning• Recall batch vs on-line (iterative) learning

- Algorithmic (statistical) approaches ~ batch- Neural-network inspired methods ~ on-line

BUT the difference is only on the implementation level (so both types of learning should yield the same generalization performance)

• Recall ERM inductive principle (for regression):

• Assume dictionary parameterization with fixed basis fcts

n

i

n

iiiiiemp fy

nyL

nR

1 1

2,1

,,1

wxwxw

ˆ y f x,w w j g j x j 1

m

46

Sequential (on-line) least squares minimization• Training pairs presented sequentially• On-line update equations for minimizing

empirical risk (MSE) wrt parameters w are:

(gradient descent learning)where the gradient is computed via the chain rule:

the learning rate is a small positive value (decreasing with k)

)(),( kykx

wxw

ww ,,1 kykLkk k

w j

L x, y, w L

ˆ y

ˆ y

w j

2 ˆ y y gj x

k

47

On-line least-squares minimization algorithm

• Known as delta-rule (Widrow and Hoff, 1960):

Given initial parameter estimates w(0), update parameters during each presentation of k-th training sample x(k),y(k)

• Step 1: forward pass computation

- estimated output• Step 2: backward pass computation

- error term (delta)

zj k gj x(k) j 1,.. .,mˆ y k wj k zj k

j 1

m

k ˆ y k y k w j k 1 w j k k k z j k , j 1,... ,m

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Neural network interpretation of delta rule• Forward pass Backward pass

1 z1 k zm k

ˆ y k

w0 k w1 k

wm k

1 z1 k zm k

k ˆ y k y k

w j k k k z j k w j k 1 w j k w j k

• Biological learning

x y

w ~ xy

w

Hebbian Rule:

Synapse

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Learning for a single neuron (delta rule):• Forward pass Backward pass

1 z1 k zm k

ˆ y k

w0 k w1 k

wm k

1 z1 k zm k

k ˆ y k y k

w j k k k z j k w j k 1 w j k w j k

• How to implement gradient-descent learning in a network of neurons?

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Backpropagation training• Minimization of

with respect to parameters (weights) W, V

• Gradient descent optimization for

where

• Careful application of gradient descent leads

leads to backpropagation algorithm

n

iiiemp yfR

1

2,, VWx

V k 1 V k k gradV L x k ,y k , V k ,w k w k 1 w k k gradw L x k , y k ,V k ,w k

k 1,... ,n,. ..

L x k , y k ,V k ,w k 1

2f x ,w,V y 2

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Backpropagation: forward passfor training input x(k), estimate predicted output

ˆ y k wj k zj k j0

m

W is m 1

zj k g a j k 1 2 m

V is d m

x1 k x2 k xd k

z1 k zm k z2 k a j k x k v j k

ky

52

Backpropagation: backward passupdate the weights by propagating the error

0 k ˆ y k y k

11 k 12 k 1m k 1 j k 0 k g aj k wj k 1

w j k 1 w j k k 0 k zj k

vij k 1 vij k k 1j k xi k

x1 k x2 k xd k

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Details of backpropagation• Sigmoid activation - picture?

• simple derivative

Poor behaviour for large t ~ saturation

• How to avoid saturation?- Proper initialization (small weights)

- Pre-scaling of inputs (zero mean, unit variance)

• Learning rate schedule (initial, final)

• Stopping rules, number of epochs

• Number of hidden units

s t 1

1 exp t s t s t 1 s t

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Additional enhancements• The problem: convergence may be very slow

for error functional with different curvatures:

• Solution: add momentum term to smooth oscillations

where and is momentum parameter

w k 1 w k k z k w k

w k w k w k 1

55

Various forms of complexity control• MLP topology ~ number of hidden units• Constraints on parameters (weights) ~

weight decay• Type of optimization algorithm (many

versions of backprop., other opt. methods)• Stopping rules• Initial conditions (initial ‘small’ weights)• So many factors make it difficult to control

complexity; usually vary 1 complexity factor while keeping all others fixed

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Toy example: regression• Data set: 25 samples generated using sine-squared

target function with Gaussian noise (st. deviation 0.1). • MLP network

(two hidden units)

underfitting

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Toy example: regression• Data set: 25 samples generated using sine-squared

target function with Gaussian noise (st. deviation 0.1). • MLP network

(10 hidden units)

near-optimal

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Backpropagation for classification• Original MLP is for regression

(as introduced above)

• For classification:

- use sigmoid output unit

- during training, use real-values 0/1 for class labels

- during operation, threshold the output of a trained MLP classifier at 0.5 to predict class labels (as in HW2)

W is m 1

1 2 m

V is d m

ˆ y w jz jj 1

m

zj g x ,v j

x1 x2 xd

z1 z2 zm

59

Toy example: classification• Data set: 250 samples ~ mixture of gaussians, where

Class 0 data has centers (-0.3, 0.7) and (0.4, 0.7), and Class 1 data has centers (-0.7, 0.3) and (0.3, 0.3).

The variance of all gaussians is 0.03. • MLP classifier

(two hidden units)

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Toy example: classification

• MLP classifier (six hidden units)

~ near optimal solution

61

MLP architectures• Supervised learning: single output

i.e., classification, regression• Supervised learning:

multiple outputs

• Unsupervised learning (data compression)

1

1

2

2

k

m

x1 x2 xd

W is m k

V v1 v2 . . .vm V is d m

for each output unit

In matrix notation:

f x,w,V wj s x v j j 1

m

w0

s x v j

F x ,W ,V s xV W

62

NetTalk (Sejnowski and Rosenberg, 1987)

One of the first successful applications of backpropagation:http://www.cnl.salk.edu/ParallelNetsPronounce/index.php

• Goal: Learning to read (English text) aloud, i.e.Learn Mapping: English text phonemesusing MLP network

• Network inputs encode 7-letter window (the 4-th letter in the middle needs to be pronounced)

• Network outputs encode phonemes (used in English)• The MLP network is trained using labeled data (both

individual words and unrestricted text)

63

NetTalk architectureInput encoding: 7x29 = 203 units

Output encoding: 26 units (phonemes) Hidden layer: 80 hidden units

64

MLP networks: summary• MLP and Projection Pursuit models have the same

mathematical parameterization but very different statistical properties:MLP model ~ sum of many basis functions of projections (basis functions are the same)PP model ~ sum of a few basis functions of projections (basis functions are adapted to data)

• Model complexity control for MLP: - may be tricky as it depends on many factors (optimization method, weight initialization, network topology)- in practice, tune just one factor (with others fixed) using resampling

NOTE: implementation of resampling may be tricky (with nonlinear optimization)

65

Regression Trees (CART) • Minimization of empirical risk (squared error)

via partitioning of the input space into regions

• Example of CART partitioning for a function of 2 inputs

f x w jI x R j j 1

m

x1

x2

R1

R2

R3

R4R5

s1

s2

s3

s4

split 1 x1 ,s1

x2 ,s2

x2 ,s3 x1 ,s4

1

2

3 4

R1

R2 R3 R4R5

66

Growing CART tree

• Recursive partitioning for estimating regions (via binary splitting)

• Initial Model ~ Region (the whole input domain) is divided into two regions and

• A split is defined by one of the inputs(k) and split point s• Optimal values of (k, s) chosen so that splitting a region

into two daughter regions minimizes empirical risk• Issues:

- efficient implementation (selection of opt. split)

- optimal tree size ~ model selection(complexity control)• Advantages and limitations

R0R1 R2

67

CART model selection• Model selection strategy

(1) Grow a large tree (subject to min leaf node size)

(2) Tree pruning by selectively merging tree nodes• The final model ~ minimizes penalized risk

where empirical risk ~ MSEnumber of leaf nodes ~ regularization parameter ~

• Note: larger smaller trees • In practice: often user-defined (splitmin in Matlab)

TRR emppen ,

T

68

Example: Boston Housing data set• Objective: to predict the value of homes in Boston • Data set ~ 506 samples total

Output: value of owner-occupied homes (in $1,000’s)Inputs: 13 variables

1. CRIM per capita crime rate by town 2. ZN proportion of residential land zoned for lots over 25,000 sq.ft. 3. INDUS proportion of non-retail business acres per town 4. CHAS Charles River dummy variable (= 1 if tract bounds river; 0 otherwise) 5. NOX nitric oxides concentration (parts per 10 million) 6. RM average number of rooms per dwelling 7. AGE proportion of owner-occupied units built prior to 1940 8. DIS weighted distances to five Boston employment centres 9. RAD index of accessibility to radial highways 10. TAX full-value property-tax rate per $10,000 11. PTRATIO pupil-teacher ratio by town 12. B 1000(Bk - 0.63)^2 where Bk is the proportion of blacks by town 13. LSTAT % lower status of the population

69

Example CART trees for Boston Housing

1.Training set: 450 samples Splitmin =100 (user-defined)

R0R1 R2

70

Example CART trees for Boston Housing

2.Training set: 450 samples Splitmin =50 (user-defined)

R0R1 R2

71

Example CART trees for Boston Housing3.Training set: 455 samples Splitmin =100 (user-defined)

Note: CART model is sensitive to training samples (vs model 1)

72

Decision Trees: summary• Advantages

- speed

- interpretability

- different types of input variables

• Limitations: sensitivity to

- correlated inputs

- affine transformations (of input variables)

- general instability of trees

• Variations: ID3 (in machine learning), linear CART

73

MARS• MARS features and improvements (over CART)

- continuous approximation (via tensor-product splines)- greedy selection of low-order basis functions- variable selection (local + global)

• MARS complexity control- lack of fit measure based on Generalized Cross Validation (GCV), i.e. MSE on the training set penalized by model complexity (tree size)

• MARS applicability- good for high- and low-D problems with a small number of low-order interactions (local or global)Interaction occurs when the effect of one variable (on the output) depends on the level of other variable

74

MARS Basis FunctionsTruncated Splines

a pair of truncated linear splines

Basic building block

xt xt

(+) (-)

xt

y

b x t

75

Tensor-Product Splines

Multivariate Splines- Tensor product

adaptive selection of knot locations- Valid knot locations

 

g x,u ,v u j x j v j

q

j 1

d

0 x1

x2

v11 v21 v31

v12

v22

v32

76

MARS Tree Structure

• Each node ~ active basis function• Basis functions estimated recursively• On each path, variables are split at most once

• Depth of tree indicates interaction level

B1 x 1

B2 x B1 x

b x1 t1

B3 x B1 x

b x1 t1

B4 x B1 x

b x2 t2

B5 x B1 x

b x2 t2

B6 x B3 x

b x3 t3

B7 x B3 x

b x3 t3

y aiBi x i1

7

77

Algorithm for MARS• Forward stepwise: search over each node to find

- split variable- split point t- coefficients (weights of basis functions)that minimize lack of fit criterion (GCV)

• Backwards stepwise: remove nodes which cause- decrease of gcv or- the smallest increase of gcv

• GCV criterionwhere

r p 1 p 2R rhl

n

Remp

hmars 1 m

78

MARS Summary

• Advantages- Provides variable subset selection - Continuous approximation- Works well for low-order interactions and

additive functions- Interpretable• Limitations

- sensitive to coordinate rotation

- problems in dealing with collinear variables

- stability of MARS modeling, for small samples

79

OUTLINE• Objectives• Methods taxonomy• Linear methods• Adaptive dictionary methods• Kernel methods and local risk minimization

- kernel methods and local risk minimization- Generalized Memory-Based Learning - Constrained Topological Mapping

• Empirical comparisons• Combining methods• Summary and discussion

80

Local Risk Minimization• Local learning (memory-based learning):

estimate a function at a single point x0

• Local risk minimization

local neighborhood functionnormalizing function

• The goal is minimization of local prediction risk over a set of and over the kernel width using only training data

R , ;x0 L y, f x , K x, x0 x0

p x ,y dxdy

K x,x0 x0 K x, x0 p x dx

f x ,

81

Practical Implementation of LRM• Sumultaneous minimization of local risk over

and over kernel width is hard • Practical methods assume fixed (constant or

linear) parameterization and then adjust only the kernel width.

Local Estimation at a point x0 :(1) Select approximating functions of fixed low

complexity, and select kernel function (i.e. gaussian or hard threshold).

(2) Select optimal kernel width, providing min estimated local risk. That is, selectively decrease training sample (near x0 ) to make an estimate.

f x ,

82

LRM and Kernel Methods • Consider minimization of Local Empirical Risk

assuming constant parameterization local average

(similarly, for local linear parameterization)• Solution to LRM leads to adaptive kernel

method, because the kernel width is adapted to data at each estimation point x0. However, adaptive selection of kernel width is hard.

Remp local 1

nK x i , x0 yi f x i , 2

i 1

n

f x ,w0 w0

f x0 w0 1

nyiK x i ,x0

i 1

n

83

Practical Selection of Kernel Width • Global Adaptive Approach: the kernel width is

estimated globally, independent of a particular estimation point.

• Global model selection for k-nn regression:For a given value of k:(1) Compute a local estimate at each input(2) Compute total empirical risk of these estimates

(3) Estimate prediction risk using (analytic) model selection criterion.

Minimize this risk through appropriate selection of k.

x iˆ y i

Remp k 1

nyi ˆ y i 2

i 1

n

84

Generalized Memory-Based Learning• For a given new input, an output is

estimated via local learning using past data

• GMBL implements locally weighted linear approximation minimizing

where the kernel

has adaptable width and scale parameters estimated via cross-validation using all data

Remp local w ,w0 1

nK xi ,x0 wx i w0 yi 2

i 1

n

K x, x ,v xk x k 2vk

2

k1

d

q

85

Constrained Topological MappingRecall applying SOMto regression problem

NonadaptiveCTM Approach:Given training data (x,y) perform1. Dimensionality reduction xz

(Apply SOM to x-values of training data)2. Apply kernel regression to estimate

y=f(z) at discrete points in z-space

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

86

Adaptive CTM Implementation• Batch implementation• Local linear modeling for each CTM unit

• Variable selection via adaptive scaling

where

• Final neighborhood width (model complexity) selected via cross-validation

Remp local w j ,w0 j 1

nK zi , j w j x i w0 j yi

i 1

n

2

c j x i v

2 v l

2 c jl xil 2

l 1

d

vl w jlj 1

b

87

OUTLINE• Objectives

• Methods taxonomy

• Linear methods

• Adaptive dictionary methods

• Kernel methods and local risk minimization

• Empirical comparisons

• Combining methods

• Summary and discussion

88

Empirical ComparisonsRef: Cherkassky et al. (1996), Comparison of adaptive

methods for function estimation from samples, IEEE Transactions on Neural Networks, 7, 969-984

• Challenge of comparisons

- who performs comparisons (experts vs general users)

- goals of comparison

- synthetic vs real-life data

- importance of experimental procedure (i.e. for model selection, double resampling etc.)

89

Example comparison studyTime series prediction (Weigend & Gershenfeld 1992)

• Performed by experts on time series 1K-100K samples long

• Lessons learned/ conclusions

- knowledge of application domain is important (simplistic black-box approaches usually fail)

- successful methods are nonlinear

- custom/manual control of method’s parameters (model selection)

90

Application of Adaptive Methods

1. Choose flexible method (parameterization)

2. Choose complexity parameter

- automatic (from data) or user-selected

3. Estimate model (from training data)

4. Estimate prediction performance(on test data)

NOTE: empirical comparison (of methods) is difficult because prediction performance depends on all factors (1) - (3), in addition to data itself

91

Example Comparison Study

• Objectives and assumptions- non-expert users

- public-domain s/w for regression methods

- manual model selection using test set; just 1 or 2 user-defined parameters

- off-line training (batch mode)

- comparison focus on methods’ parameterization (1) and model selection (2) for different synthetic data sets

92

Objectives and assumptions (cont’d)• Methods in XTAL package

k- nearest neighbors regression (k-NN)Linear Regression (LR)Projection Pursuit (PPR)Multivariate Adaptive Regression Splines (MARS)Generalized Memory-Based Learning (GMBL)Constrained Topological Mapping (CTM)Artificial Neural Network / backpropagation (ANN)

• Synthetic datalow- and high-dimensionaluniformly distributed in x-space

93

Experimental Set-Up• Specification of

properties of synthetic data: target functions, training/test set size, x-distribution, noise level

performance metric: NRMS error (for test set)4 parameter settings for each method:

KNN: k = 2, 4, 8, 16

GMBL: no parameters (run only once)

CTM: smoothing parameter = 0, 2, 5, 9

MARS: smoothing parameter = 0, 2, 5, 9

PPR: number of terms (in the smallest model) = 1, 2, 5, 8

ANN: number of hidden units = 5, 10, 20, 40

94

• Training Data- uniform distribution (random and spiral)

- size: smalle (25), medium (100), large (400)

- noise level: none, medium(SNR=4), large (SNR=2)

• Test data: 961 samples (no noise)- spaced uniformly on 2D grid (for 2-dimensional data)

- randomly sampled for high-dimensional data

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

X1

X2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

X1

X2

95

• 2D Target functions

Function 3 Function 4

Function 1 Function 2

96

• 2D Target functions (cont’d)

Function 5 Function 6

Function 7 Function 8

97

Comparison SummaryBEST WORST

Prediction accuracy (dense samples) ANN KNN, GMBLPrediction accuracy (sparse samples)GMBL, KNN MARS, PPAdditive target functions MARS, PP KNN, GMBLHarmonic functions CTM, ANN PPRadial functions ANN, PP KNNRobustness wrt parameter tuning ANN, GMBL PPRobustness wrt sample properties ANN, GMBL PP, MARS

• Methods performance- similar at dense (large) samples- uneven at sparse samples and depends significantly on the properties of data

98

Comments on Specific Methods

• Comparison metrics:

(a) generalization

(b) robust parameter tuning

(c)robust to data characteristics

• kNN and GMBL: (a) inferior to other methods when accurate prediction is possible

(b) very robust

(c) very robust

99

Comments on Specific Methods

• MARS(a) good for additive functions(b) somewhat brittle(c) rather unpredictable

• PPR (a) good for additive functions, functions of linear combinations of inputs; poor for harmonic functions(b) brittle(c) rather unpredictable

100

Comments on Specific Methods

• ANN(a) good for functions of linear combinations, harmonic and radial-type functions(b) very robust(c) very predictable

• CTM (a) very good for harmonic functions, poor functions of linear combinations(b) robust(c)predictable; best for spiral distribution in x-space

101

Conclusions and Caveats• Comparison results always biased by

- selection of data sets- s/w implementation of adaptive methods- (expert) user bias

• Relative performance varies with properties of data sets (i.e. sample size, noise level etc)

• Heuristic optimization methods (ANN, CTM) are computationally intensive but often more robust than faster statistical methods

• Nonlinear methods should be robust: only for robust methods it is possible to develop automatic parameter tuning (complexity control)

102

OUTLINE• Objectives

• Methods taxonomy

• Linear methods

• Adaptive dictionary methods

• Kernel methods and local risk minimization

• Empirical comparisons

• Combining methods

• Summary and discussion

Motivation for Combining Methods

• General setting (used in this course)

- given training data set

- apply different learning methods

- select the best model (method)

Learning Method + Data Predictive Model

• Why discard other models?

Motivation (cont’d)

Learning Method + Data Predictive Model• Theoretical and empirical evidence

- no single ‘best’ method exists• Always possible to find:

- best method for given data set- best data set for given method

• Philosophical + statistical connections, Eastern philosophy, Bayesian averaging:

Combine several theories (models) explaining the data

Strategies for Combining Methods

• Predictive model depends on 3 factors (a) parameterization of admissible models(b) random training sample(c) empirical loss (for risk minimization)

• Three combining strategies (for improved generalization)

1. Different (a), the same (b) and (c) Committee of Networks, Stacking, Bayesian averaging2. Different (b), the same (a) and (c) Bagging3. Different (c), the same (a) and (b) Boosting

Combining Strategy 1• Apply N different methods (parameterizations) to

the same data N distinct models• Form (linear) combination of N models

Combining Strategy 1 (cont’d)

Design issues:• What parameterizations (methods) to use?

- as different as possible

• How many component models?

• How to combine component models?

- via empirical risk minimization (neural network strategy)

- Bayesian averaging (statistical strategy)

Committee of networks approachGiven training data • Estimate N candidate (regression) models

using different methods• Construct the combined model as

where coefficients are estimated via min. of emp. riskunderconstraints

x i ,yi i 1, . . . ,n

**22

*11 ,,...,,,, NNfff xxx

N

jjjjcom f

Nf

1

*,1

, xx

n

iiicom y

nR

1

21 ,f x 11

N

jj

j 0

j

Example of Committee Approach• Regression data set:

with x-values uniform in [0,1] and noise variance

• Regression methods used(a) polynomial(b) trigonometric(c) combined (Committee of Networks)

• Model selection:

- VC model selection for (a) and (b)- empirical risk minimization for (c)

y 0.8sin 2 x 0 .2x 2

2 0 .25

f trig x ,vm2, wm2

v j sin jx w j cos jx j 1

m2 1

w0

f comb1 x , fpoly x,um1 1 f trig x,vm2

,wm 2

f poly x ,um1 uj x

j

j 0

m1 1

Comparison: 25 training samplesRed ~ target function; Blue dashed ~ polynomial modelBlue dotted ~ trigonometric model; Black ~ combined model

Comparison: 50 training samplesRed ~ target function; Blue dashed ~ polynomial modelBlue dotted ~ trigonometric model; Black ~ combined model

Comparison results

25 training samples: Model f(x) MSE(f(x),target) Poly (d=3) 0.0857

Trigon (d=3) 0.0237

Combined 0.0239

(alpha=0.5)

50 training samples: Model f(x) MSE(f(x),target) Poly (d=4) 0.0046

Trigon (d=4) 0.0044

Combined 0.0038

(alpha=0.2)

Stacking approachGiven training data • Estimate N candidate (regression) models

using different methods• Construct the combined model as

where coefficients are estimated via resamplingunderconstraints

x i ,yi i 1, . . . ,n

**22

*11 ,,...,,,, NNfff xxx

N

jjjjcom f

Nf

1

*,1

, xx

n

iiicom y

nR

1

21 ,f x 11

N

jj

j 0

j

114

Empirical Comparison• The same data set and experimental setup• Committee approach ~ Comb 1

Stacking approach ~ Comb 2

0.26

0.28

0.3

0.32

0.34

0.36

poly trig comb1 comb2 k-nn

method

Risk, noise variance 0.25, n=50

115

Summary and Discussion• Linear (nonadaptive) methods for regression

- theoretically well-understood

- effective methods for complexity control• Nonlinear (adaptive) methods

- inherently complex (non-tractable optimization)

- difficult to apply analytic model selection and resampling

- no single best method exists for all data sets• Combining methods often result in better

predictions