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Instance Based Learning

Ata KabanThe University of Birmingham

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Today we learn: K-Nearest Neighbours Locally weighted regression Case-based reasoning Lazy and eager learning

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Instance-based learning

One way of solving tasks of approximating discrete or real valued target functions

Have training examples: (xn, f(xn)), n=1..N. Key idea:

– just store the training examples– when a test example is given then find the

closest matches

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1-Nearest neighbour:Given a query instance xq,

• first locate the nearest training example xn

• then f(xq):= f(xn) K-Nearest neighbour:

Given a query instance xq, • first locate the k nearest training examples • if discrete values target function then take

vote among its k nearest nbrs else if real valued target fct then take the mean of the f values of the k nearest nbrs

kxf

xfk

i iq

1)(

:)(

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The distance between examples

We need a measure of distance in order to know who are the neighbours

Assume that we have T attributes for the learning problem. Then one example point x has elements xt , t=1,…T.

The distance between two points xi xj is often defined as the Euclidean distance:

T

ttjtiji xxd

1

2][),( xx

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Voronoi Diagram

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Characteristics of Inst-b-Learning

An instance-based learner is a lazy-learner and does all the work when the test example is presented. This is opposed to so-called eager-learners, which build a parameterised compact model of the target.

It produces local approximation to the target function (different with each test instance)

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When to consider Nearest Neighbour algorithms?

Instances map to points in Not more then say 20 attributes per

instance Lots of training data Advantages:

– Training is very fast– Can learn complex target functions– Don’t lose information

Disadvantages:– ? (will see them shortly…)

n

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twoone

four

three

five six

seven Eight ?

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Training dataNumber Lines Line types Rectangles Colours Mondrian?

1 6 1 10 4 No

2 4 2 8 5 No

3 5 2 7 4 Yes

4 5 1 8 4 Yes

5 5 1 10 5 No

6 6 1 8 6 Yes

7 7 1 14 5 No

Number Lines Line types Rectangles Colours Mondrian?

8 7 2 9 4

Test instance

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Keep data in normalised formOne way to normalise the data ar(x) to a´r(x) is

t

ttt

xxx

'

attributestofmeanx thr

attributestofdeviationndardsta tht

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Normalised training dataNumber Lines Line

types Rectangles Colours Mondrian?

1 0.632 -0.632 0.327 -1.021 No

2 -1.581 1.581 -0.588 0.408 No

3 -0.474 1.581 -1.046 -1.021 Yes

4 -0.474 -0.632 -0.588 -1.021 Yes

5 -0.474 -0.632 0.327 0.408 No

6 0.632 -0.632 -0.588 1.837 Yes

7 1.739 -0.632 2.157 0.408 No

Number Lines Line types

Rectangles Colours Mondrian?

8 1.739 1.581 -0.131 -1.021

Test instance

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Distances of test instance from training data

Example Distanceof testfromexample

Mondrian?

1 2.517 No

2 3.644 No

3 2.395 Yes

4 3.164 Yes

5 3.472 No

6 3.808 Yes

7 3.490 No

Classification1-NN Yes

3-NN Yes

5-NN No

7-NN No

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What if the target function is real valued?

The k-nearest neighbour algorithm would just calculate the mean of the k nearest neighbours

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Variant of kNN: Distance-Weighted kNN

We might want to weight nearer neighbors more heavily

Then it makes sense to use all training examples instead of just k (Stepard’s method)

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1

1

),(1 where

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iqik

i i

k

i iiq d

ww

fwf

xxx

x

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Difficulties with k-nearest neighbour algorithms

Have to calculate the distance of the test case from all training cases

There may be irrelevant attributes amongst the attributes – curse of dimensionality

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A generalisation: Locally Weighted Regression

Some useful terminology– Regression = approximating a real valued

function– Residual error = the error in approximating

the target function– Kernel function = the function of distance

that is used to determine the weight of each training example, I.e.

)),((:),(

12 iq

iqi xxdK

xxdw

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(Locally Weighted Regression)

Note kNN forms local approximations to f for each query point Why not form an explicit local approximation for regions

surrounding the query point, e.g.– Fit a linear function to the k nearest neighbours (or a quadratic, …) e.g.

– …

)(

2

110

))(ˆ)((21)( minimize

)(...)()(ˆlet

qxkNNxq

nn

xfxfxE

xawxawwxf

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Case-based reasoning (CBR) CBR is an advanced instance based learning

applied to more complex instance objects Objects may include complex structural

descriptions of cases & adaptation rules It doesn’t use Euclidean distance measures but

can do matching between objects using e.g. semantic nets.

It tries to model human problem-solving– uses past experience (cases) to solve new problems– retains solutions to new problems

CBR is an ongoing area of machine learning research with many applications

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Applications of CBR Design

– landscape, building, mechanical, conceptual design of aircraft sub-systems

Planning– repair schedules

Diagnosis– medical

Adversarial reasoning– legal

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CBR processNew Case

matching Matched Cases

Retrieve

Adapt?No

Yes

Closest CaseSuggest solution

Retain

Learn

Revise

Reuse

Case Base

Knowledge and Adaptation rules

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CBR example: Property pricingCase Location

codeBedrooms Recep

roomsType floors Cond-

itionPrice£

1 8 2 1 terraced 1 poor 20,500

2 8 2 2 terraced 1 fair 25,000

3 5 1 2 semi 2 good 48,000

4 5 1 2 terraced 2 good 41,000

Case Locationcode

Bedrooms Receprooms

Type floors Cond-ition

Price£

5 7 2 2 semi 1 poor ???

Test instance

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How rules are generated

Examine cases and look for ones that are almost identical– case 1 and case 2

• R1: If recep-rooms changes from 2 to 1 then reduce price by £5,000

– case 3 and case 4• R2: If Type changes from semi to terraced then

reduce price by £7,000

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Matching Comparing test instance

– matches(5,1) = 3– matches(5,2) = 3– matches(5,3) = 2– matches(5,4) = 1

Estimate price of case 5 is £25,000

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Adapting

Reverse rule 2– if type changes from terraced to semi then

increase price by £7,000 Apply reversed rule 2

– new estimate of price of property 5 is £32,000

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Learning

So far we have a new case and an estimated price– nothing is added yet to the case base

If later we find house sold for £35,000 then the case would be added– could add a new rule

• if location changes from 8 to 7 increase price by £3,000

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Problems with CBR

How should cases be represented? How should cases be indexed for fast

retrieval? How can good adaptation heuristics be

developed? When should old cases be removed?

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Advantages

An local approximation is found for each test case

Knowledge is in a form understandable to human beings

Fast to train

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Summary

K-Nearest Neighbour (Locally weighted regression) Case-based reasoning Lazy and eager learning

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Lazy and Eager Learning

Lazy: wait for query before generalizing– k-Nearest Neighbour, Case based reasoning

Eager: generalize before seeing query– Radial Basis Function Networks, ID3, …

Does it matter?– Eager learner must create global approximation– Lazy learner can create many local

approximations