Association rules and compositional data analysis: implications to big dataR. 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, Israel2Universitat de Girona, Spain

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This is work in progress

The long term goal is to introduceCoDa 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 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 otherLift > 1, A and B have affinity to each other

OR {A=>B} = (x1*x4)/(x2*x3)

OR < 1, A and B repel each otherOR > 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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0D

M

DRLD

D

MDD

4 2 3D 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 0 1 2-1

.5-0

.50.

51.

5

ilr1

ilr2

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

clr(𝒙) = (log(𝑥1

𝑔 𝑥), log(

𝑥2

𝑔 𝑥),..., log(

𝑥𝐷

𝑔 𝑥))

ilr(𝒙)=(ilr1(𝒙), … , ilrD−1(𝒙))

Simplex: raw data (%)

𝒙 = (𝑥1, 𝑥2,..., 𝑥𝐷)

x1 x2

x3

x1

x3

x2

3

Logratio (multiplicative) approach

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CoDa Analysis of 2X2 tables T

𝑖𝑙𝑟 𝐓 =1

2ln

𝑥1𝑥4𝑥2𝑥3

,2

2ln

𝑥1𝑥4

,2

2ln

𝑥2𝑥3

.

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 2X2 tables

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ilr-coordinates ilr1 ilr2 ilr3

T1

2ln

𝑥1𝑥4𝑥2𝑥3

2

2ln

𝑥1𝑥4

2

2ln

𝑥2𝑥3

Tind 02

2ln

𝑥1𝑥4

2

2ln

𝑥2𝑥3

Tint

1

2ln

𝑥1𝑥4𝑥2𝑥3

0 0

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 2X2 tables

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X f g De e

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

Let 𝐓 𝑎 = 𝑖𝑙𝑟(𝐱) be the Aitchison norm of a table T,

then 𝐓 𝑎2 = 𝐓𝑖𝑛𝑑 𝑎

2 + 𝐓𝑖𝑛𝑡 𝑎2 , that is, one has a

decomposition of the Aitchison norm of table T.

independence interaction

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CoDa Simplical Deviance (SD)

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𝑆𝐷(𝐓) = 𝐓𝑖𝑛𝑡 𝑎2 =

1

4l𝑛2

𝑥1𝑥4

𝑥2𝑥3= 𝑖𝑙𝑟1

2(𝐓)

𝑥1𝑥4𝑥2𝑥3

= 1 ⟺ 𝑙𝑛𝑥1𝑥4𝑥2𝑥3

= 0 ⟺ 𝑖𝑙𝑟1 𝐓 = 0 ⟺ 𝑆𝐷 = 0

X f g De e independence interaction

CoDa Relative Simplical Deviance (SD)

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𝑅𝑆𝐷(𝐓) =𝑆𝐷

𝐓 𝑎2 =

)𝑖𝑙𝑟12(𝐓

)𝑖𝑙𝑟(𝐓 2M

DRLD

D

3 2

3

2

0If Dthenif x x

Dthen RLD

D xD

else RLDD x

1 4

1

4

elseif x x

Dthen RLD

D xD

else RLDD 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 followingsteps: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) =𝑥1

𝑥1 + 𝑥2)(𝑥1 + 𝑥3

𝐷(AR) = 𝑥1𝑥4 − 𝑥2𝑥3

lift AR = 1 +)𝐷(AR

𝑥1 + 𝑥2)(𝑥1 + 𝑥3

CoDa Measures for Association Rules

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𝑂𝑅∗ AR = 𝑌𝑢𝑙𝑒′𝑠 𝑄 𝐴𝑅 =𝑥1𝑥4−𝑥2𝑥3

𝑥1𝑥4+𝑥2𝑥3

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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𝐶 AR = 𝑖𝑙𝑟1 𝐓

𝐶∗ AR = tanh 𝐶 AR = 𝑂𝑅∗ AR = 𝑌𝑢𝑙𝑒′𝑠 𝑄 𝐴𝑅

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

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CoDa Measures for Association Rules - 𝐶(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 Visualizationilr plot

by consequent item

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interaction

ilr.1

fre

qu

en

cy

0.2 0.4 0.6 0.8 1.0

05

10

15

CoDa AR Visualizationilr plot

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interaction

CoDa AR Visualizationclr plot

by consequent 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