Association rules and compositional data analy lysis: : im impli - - PowerPoint PPT Presentation

Association rules and compositional data analy lysis: : im impli licatio ions to big ig data

• R. S. Kenett1, J.A. Martín-Fernández2, S. Thió-Henestrosa2 and M. Vives-Mestres2

1 KPA Group, Israel; University of Turin, Italy and Neaman Institute, Technion, Israel 2Universitat de Girona, Spain

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This is work in progress The long term goal is to introduce CoDa to text (semantic data) analysis and to scale it to big data….

Association Rules(AR)

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Transaction (document, itemset)

LHS (A) RHS (B)

Antecedent Consequent

Basket Analysis

Terms, items, tokens, words

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AR: : Support, Confidence, , Lift ft and Odd Ratio

RHS ^RHS LHS x1 x2 g ^LHS x3 x4 1-g f 1-f 1

support {A=>B} = x1

A B

4 1

1, 0 , 1...4.

i i i

x x i



    



Proportion of transactions in which an item set appears

confidence {A=>B} = x1/g

Strength of implication, or predictive power

lift {A=>B} = confidence{A=>B} / support{B} = support{A=>B}/support{A}support{B} Lift < 1, A and B repel each other Lift > 1, A and B have affinity to each other OR {A=>B} = (x1*x4)/(x2*x3) OR < 1, A and B repel each other OR > 1, A and B have affinity to each other

The Simplex

4 1

1, 0 , 1...4.

i i i

x x i



    



RHS ^RHS LHS x1 x2 ^LHS x3 x4

A B

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D 

M

D RLD D 

M

D

D

4 2 3

D x x x x



 

Relative Linkage Disequilibrium

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lift {A=>B} = 1 OR {A=>B} = 1 1 2 3 4

( , , , ) ( ,1 ) ( ,1 ) (1, 1) where X x x x x f f f g g g e       

X f g De e    

RHS ^RHS LHS x1 x2 g ^LHS x3 x4 1-g f 1-f 1 Kenett, R.S. (1983). On an Exploratory Analysis of Contingency

• Tables. J R Stat Soc Series D, 32, 395—403.

Kenett R.S. (2014). Frequenct vectors and contingency tables: a non paramtric and graphical analysis. Girona Seminar, 27/11/14.

independence dependence

CoDa Analysis and Principles

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• Multiplicative tools to CoDa are equivalent to classical

additive (Euclidean) tools to log-ratio values

• Transform CoDa, e.g. isometric log-ratio coordinates: ilr(x)
• Scale invariance (Vectors P =

[p1,…,pD] and P’ = αP, α > 0, give the same information

• Subcompositional coherence

• 2
• 1

1 2

• 1.5
• 0.5

0.5 1.5 ilr1 ilr2

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

Real space: log-ratio coordinates (alr, clr, or ilr)

clr(𝒚) = (log(

𝑦1 𝑕 𝑦 ), log( 𝑦2 𝑕 𝑦 ),..., log( 𝑦𝐸 𝑕 𝑦 ))

ilr(𝒚)=(ilr1(𝒚), … , ilrD−1(𝒚))

Simplex: raw data (%)

𝒚 = (𝑦1, 𝑦2,..., 𝑦𝐸)

x1 x2 x3

x1 x3 x2 3

Logratio (multiplicative) approach

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CoDa Analysis of f 2X2 2 tables T

𝑗𝑚𝑠 𝐔 = 1 2 ln 𝑦1𝑦4 𝑦2𝑦3 , 2 2 ln 𝑦1 𝑦4 , 2 2 ln 𝑦2 𝑦3 .

Sequential Binary Partition (SBP), Pawlowsky-Glahn and Buccianti, 2011, Chapter 2.

RHS ^RHS LHS x1 x2 g ^LHS x3 x4 1-g f 1-f 1

A B

4 1

1, 0 , 1...4.

i i i

x x i



    



CoDa Analysis of f 2X2 2 tables

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ilr-coordinates ilr1 ilr2 ilr3 T 1 2 ln 𝑦1𝑦4 𝑦2𝑦3 2 2 ln 𝑦1 𝑦4 2 2 ln 𝑦2 𝑦3 Tind 2 2 ln 𝑦1 𝑦4 2 2 ln 𝑦2 𝑦3 Tint 1 2 ln 𝑦1𝑦4 𝑦2𝑦3

independence interaction Perturbation operation subtracting table Tind from T

ilr1(T) < 0 : negative effect between itemsets (A true, B less likely true) ilr1(T) = 0 : independence ilr1(T) > 0 : positive effect (A true, B more likely true)

CoDa Analysis of f 2X2 2 tables

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X f g De e    

ilr(T)=ilr(Tind)+ilr(Tint).

Let 𝐔 𝑏 = 𝑗𝑚𝑠(𝐲) be the Aitchison norm of a table T, then 𝐔 𝑏

2 =

𝐔𝑗𝑜𝑒

𝑏 2 + 𝐔𝑗𝑜𝑢 𝑏 2 , that is, one has a

decomposition of the Aitchison norm of table T.

independence interaction

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CoDa Simplical Deviance (S (SD)

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𝑇𝐸(𝐔) = 𝐔𝑗𝑜𝑢

𝑏 2 = 1 4 l𝑜2 𝑦1𝑦4 𝑦2𝑦3 = 𝑗𝑚𝑠 1 2(𝐔)

𝑦1𝑦4 𝑦2𝑦3 = 1 ⟺ 𝑚𝑜 𝑦1𝑦4 𝑦2𝑦3 = 0 ⟺ 𝑗𝑚𝑠

1 𝐔 = 0 ⟺ 𝑇𝐸 = 0

X f g De e    

independence interaction

CoDa Relative Simplical Deviance (S (SD)

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𝑆𝑇𝐸(𝐔) = 𝑇𝐸 𝐔 𝑏

2 =

) 𝑗𝑚𝑠

1 2(𝐔

) 𝑗𝑚𝑠(𝐔

2

M

D RLD D 

3 2 3 2

If D then if x x D then RLD D x D else RLD D x            

1 4 1 4

else if x x D then RLD D x D else RLD D x          

RSD takes values in an interval [0,1]

Bootstrap Algorithm

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Egozcue et al. (2015) introduce a bootstrap algorithm consisting of following steps: i) Calculate Tind, Tint, SD and RSD. ii) Simulate 10000 multinomial samples (T(k)) assuming the independence hypothesis H0: T=Tind is true. For each table T(k), calculate T(k)

ind, T(k) int, SD(k)

and RSD(k). iii) Compare respectively the value of SD and RSD with the distribution of the 10000 values of SD(k) and RSD(k) to obtain the percentile p-value (left tail). Calculate the 0.05 significance critical points (5th quantile) in the left tail of each distribution.

CoDa Measures for Association Rules

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lift(AR) = 𝑦1 𝑦1 + 𝑦2)(𝑦1 + 𝑦3

𝐸(AR) = 𝑦1𝑦4 − 𝑦2𝑦3 lift AR = 1 + ) 𝐸(AR 𝑦1 + 𝑦2)(𝑦1 + 𝑦3

CoDa Measures for Association Rules

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𝑃𝑆∗ AR = 𝑍𝑣𝑚𝑓′𝑡 𝑅 𝐵𝑆 = 𝑦1𝑦4−𝑦2𝑦3

𝑦1𝑦4+𝑦2𝑦3

OR(AR) = odds(B/A)/odds(B/cA) = (x1x4)/(x2x3). Lift(AR) =1 D(AR) =0 OR(AR) =1

OR is defined in [0, +Infinite), OR* is defined in [-1,1]

CoDa Measures for Association Rules

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𝐷 AR = 𝑗𝑚𝑠

1

𝐔

𝐷∗ AR = tanh 𝐷 AR = 𝑃𝑆∗ AR = 𝑍𝑣𝑚𝑓′𝑡 𝑅 𝐵𝑆

C is defined in (-Infinite, +Infinite), C* is defined in [-1,1]

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CoDa Measures for Association Rules - 𝐷(A (AR)

A Case Study

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https://treato.com

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https://treato.com/Nicardipine/?a=s

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Document Term Matrix (DTM)

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Association Rules Analysis

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Association Rules by consequent (lowest Lift)

Vasoconstriction

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Top 10 Association Rules by Lift (in red)

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Top 10 Association Rules by Lift (in red)

CoDa AR Vis isualization ilr ilr plo lot by consequent it item

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interaction

CoDa AR Visualization ilr ilr plot

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interaction

CoDa AR Vis isualization clr lr plo lot by consequent it item

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interaction

Conclusions

• Compositional measures of independence SD and RSD are coherent with

the simplicial geometry of the simplex, the sample space of contingency tables of AR.

• The relation between CoDa-AR measures and other common measures

facilitates the interpretation of negative and positive effects between itemsets.

• The CoDa geometry provides visualization techniques of measures when all

the significant AR of a large database are analyzed.

• The principles of coherence and scalability, that are fundamental to CoDa,

are relevant to big data text analysis.

• More research in this area is needed

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Acknowledgements

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https://www.kaggle.com/c/instacart-market-basket-analysis

Thank you for your attention