Estimate the Number of Estimate the Number of Relevant Images Using Two-Relevant Images Using Two-

Order Markov Chain Order Markov Chain

Presented by: WANG XiaolingPresented by: WANG Xiaoling

Supervisor: Clement LEUNGSupervisor: Clement LEUNG

OutlineOutline

IntroductionIntroduction Objective Objective MethodologyMethodology Experiment ResultsExperiment Results Conclusion and Future WorkConclusion and Future Work

IntroductionIntroduction

Large collections of images have Large collections of images have been made available on web.been made available on web.

Retrieval effectiveness becomes one Retrieval effectiveness becomes one of the most important parameters to of the most important parameters to measure the performance of image measure the performance of image retrieval systems. retrieval systems.

Measures: Measures: PrecisionPrecision Recall Recall

Significant Challenge: the total number of Significant Challenge: the total number of relevant images is relevant images is not directly observablenot directly observable

databasetheinimagesrelevantofnumbertotal

retrievedimagesrelevantofnumberR

returnedimagesofnumbertotal

retrievedimagesrelevantofnumberP

Basic ModelsBasic Models Regression ModelRegression Model Markov ChainMarkov Chain Two-Order Markov ChainTwo-Order Markov Chain

ObjectiveObjective

To investigate the probabilistic To investigate the probabilistic behavior of the distribution of behavior of the distribution of relevant images among the returned relevant images among the returned results for the image search engines results for the image search engines using two-order markov chainusing two-order markov chain

MethodologyMethodology

Test Image Search Engine: Test Image Search Engine: Query DesignQuery Design

70% provided by authors70% provided by authors One word queryOne word query Two word queryTwo word query Three word queryThree word query

30% suggestive term30% suggestive term Suggestive term with largest returned resultsSuggestive term with largest returned results Suggestive term with least returned resultsSuggestive term with least returned results

MethodologyMethodology

Database Setup:Database Setup: Stochastic process {Stochastic process {XX11, X, X22,…,,…, XXJJ } }

where where XXJJ denotes the aggregate relevanc denotes the aggregate relevance of all the images in page e of all the images in page J J Equation:Equation:

where where YYJiJi=1=1 if the if the i i th th image on page image on page JJ is re is relevant, andlevant, and Y YJiJi =0 if the =0 if the i i thth image on page image on page JJ is not relevant. is not relevant.

20

1J YX

iJi

Page Page JJ XXJJ

11 1818

22 1919

33 2020

44 1919

55 2020

66 1919

77 2020

88 1818

99 1919

1010 1818

18YX20

1J

iJi

Forecast Using Two-Order Markov ChainForecast Using Two-Order Markov Chain Markov Chain: Stochastic process {XMarkov Chain: Stochastic process {XJJ, J, J≥1≥1} with state s} with state s

pace S={0,1,2,…20} pace S={0,1,2,…20} ，， Two-Order Markov Chain: State space change to STwo-Order Markov Chain: State space change to S22 ，， Forecast the state probability distribution of next page Forecast the state probability distribution of next page

ππ(J) (J) based on the original state probability distributiobased on the original state probability distribution n ππ(1) (1) and transition probability matrix and transition probability matrix PP . . An ExampleAn Example

Model TestModel Test Mean Absolute ErrorMean Absolute Error

Experiment ResultsExperiment Results Forecast Results Using Two-Order Markov ChainForecast Results Using Two-Order Markov Chain

PagePage GoogleGoogle YahooYahoo BingBing

11 2020 2020 2020

22 2020 2020 2020

33 2020 2020 2020

44 2020 2020 2020

55 2020 2020 2020

66 2020 2020 1717

77 2020 2020 1717

88 2020 2020 1717

99 2020 2020 1717

1010 2020 2020 1717

Test Results--GoogleTest Results--Google

1 2 3 4 5 6 7 8 9 1010

15

20

Page Number

( i )

1 2 3 4 5 6 7 8 9 1010

15

20

Page Number

( j )

1 2 3 4 5 6 7 8 9 1010

15

20

Page Number

( a )

1 2 3 4 5 6 7 8 9 1010

15

20

Page Number

( b )

1 2 3 4 5 6 7 8 9 1010

15

20

Page Number

Num

ber o

f Rel

evan

t Im

ages ( c )

1 2 3 4 5 6 7 8 9 1010

15

20

Page Number

Num

ber o

f Rel

evan

t Im

ages ( d )

1 2 3 4 5 6 7 8 9 1010

15

20

Page Number

( e )

1 2 3 4 5 6 7 8 9 1010

15

20

Page Number

( f )

1 2 3 4 5 6 7 8 9 1010

15

20

Page Number

( g )

1 2 3 4 5 6 7 8 9 1010

15

20

Page Number

( h )

Test Results--YahooTest Results--Yahoo

1 2 3 4 5 6 7 8 9 1010

15

20

( i )

Page Number1 2 3 4 5 6 7 8 9 10

10

15

20

( j )

Page Number

1 2 3 4 5 6 7 8 9 1010

15

20( a )

Page Number1 2 3 4 5 6 7 8 9 10

10

15

20( b )

Page Number

1 2 3 4 5 6 7 8 9 1010

15

20( c )

Page Number

Num

ber o

f Rel

evan

t Im

ages

1 2 3 4 5 6 7 8 9 1010

15

20( d )

Page Number

Num

ber o

f Rel

evan

t Im

ages

1 2 3 4 5 6 7 8 9 1010

15

20( e )

Page Number1 2 3 4 5 6 7 8 9 10

10

15

20( f )

Page Number

1 2 3 4 5 6 7 8 9 1010

15

20( g )

Page Number1 2 3 4 5 6 7 8 9 10

10

15

20( h )

Page Number

Test Results--BingTest Results--Bing1 2 3 4 5 6 7 8 9 10

1 2 3 4 5 6 7 8 9 100

10

20(c)

Page Number

Num

ber o

f Rel

evan

t Im

ages

1 2 3 4 5 6 7 8 9 100

10

20(e)

Page Number

1 2 3 4 5 6 7 8 9 100

10

20(g)

Page Number

1 2 3 4 5 6 7 8 9 100

10

20(a)

Page Number

1 2 3 4 5 6 7 8 9 100

10

20

(i)

Page Number1 2 3 4 5 6 7 8 9 10

0

10

20

(j)

Page Number

1 2 3 4 5 6 7 8 9 100

10

20(b)

Page Number

1 2 3 4 5 6 7 8 9 100

10

20(d)

Page Number

Num

ber o

f Rel

evan

t Im

ages

1 2 3 4 5 6 7 8 9 100

10

20(f)

Page Number

1 2 3 4 5 6 7 8 9 100

10

20(h)

Page Number

Measure for Forecast AccuracyMeasure for Forecast Accuracy

MMeanean A Absolute bsolute DDeviationeviation ( (MADMAD)) ::

n

errorforecast MAD

One-wordOne-word Two-wordTwo-word Three-wordThree-word

GooglGooglee

2.2.77

2.2.33

1.11.1 0.80.8 0.10.1 0.80.8 1.1.77

1.91.9 0.60.6

YahooYahoo 2.2.00

0.0.11

1.11.1 0.50.5 4.84.8 1.11.1 2.2.22

2.22.2 0.40.4

BingBing 1.1.33

1.1.99

1.21.2 4.54.5 2.02.0 1.21.2 1.1.22

10.10.55

1.11.1

Comparative ResultsComparative Results

Best Model: Two-Best Model: Two-Order Markov Order Markov ChainChain

Worst Model: Worst Model: Regression ModelRegression Model

0

1

2

3

4

5

Googel Yahoo Bi ng

Image Search Engi ne

Mean

Abs

olut

e Er

ror

Regreesi onModelMarkov Chai n

Two-OrderMarkov Chai n

ConclusionConclusion

Two-Order Markov Chain could well Two-Order Markov Chain could well represent the distribution of relevant represent the distribution of relevant images among the results pages for images among the results pages for the major web image search engine.the major web image search engine.

Two-Order Markov Chain is the best Two-Order Markov Chain is the best model among three models we have model among three models we have worked.worked.

Future WorkFuture Work

Our future work will try to apply Hidden Our future work will try to apply Hidden Markov Chain to this topicMarkov Chain to this topic

Thank you!Thank you!

Q & AQ & A

Two-Order Markov Chain Two-Order Markov Chain An example (cont’)An example (cont’)

Suppose the stochastic process {XSuppose the stochastic process {Xtt, t>=0} with , t>=0} with a state space S={A, B, C}a state space S={A, B, C}

As to two-order Markov chain, the state space:As to two-order Markov chain, the state space: SS22={AA, AB, AC, BA, BB, BC, CA, CB, CC} ={AA, AB, AC, BA, BB, BC, CA, CB, CC} The state probabilities distribution of period zThe state probabilities distribution of period z

ero: ero: (0)= ((0)= (AAAA, , ABAB, , ACAC, , BABA, , BBBB, , BCBC, , CACA, , CBCB, , CCCC))

An example (cont’)An example (cont’)

The transition probability matrix:The transition probability matrix:

CCCCCBCCCACC

BCCBBBCBBACB

ACCAABCABCCA

CCBCCBBCCABC

BCBBBBBBBABB

ACBAABBAAABA

CCACCBACCAAC

BCABBBABBAAB

ACAAABAAAAAA

ppp

ppp

ppp

ppp

ppp

ppp

ppp

ppp

ppp

p

,,,

,,,

,,,

,,,

,,,

,,,

,,,

,,,

,,,

000000

000000

000000

000000

000000

000000

000000

000000

000000

PAA,BA=0

An exampleAn example

Therefore, the probability distribution of Therefore, the probability distribution of states for page states for page J J will be compute as:will be compute as:

ππ(J)=(J)=ππ(J-1)*P(J-1)*P

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