Multiway-SIR for Longitudinal Multi-table Data Integration

(1)

Multiway-SIR for Longitudinal Multi-table Data Integration

Nathalie Villa-Vialaneix, Valérie Sautron, Marie Chavent & Nathalie Viguerie

[email protected] http://www.nathalievilla.org

COMPSTAT - August 25th, 2016

Nathalie Villa-Vialaneix |SIR-STATIS 1/24

(2)

Sommaire

1 Background and motivation

2 Presentation of Multiway-SIR

3 Application

(3)

Sommaire

3 Application

(4)

A longitudinal study on weight loss induced by a low calorie diet

Data collected during EU project, 8 centers, 450 families Purpose: effect of glycemic index, protein content... on weight maintenance after a diet for obese people

Data description: at each Clinical Intervention Day (CID), measure of 15 clinical variables (weight, HDL, ...) on 135 obese women

Targeted problem: exploratory data analysis aimed at explaining the success/failure of the diet (in term of weight loss/regain)

(5)

A longitudinal study on weight loss induced by a low calorie diet

Targeted problem: exploratory data analysis aimed at explaining the success/failure of the diet (in term of weight loss/regain)

(6)

A longitudinal study on weight loss induced by a low calorie diet

(7)

Data description and notations

X_..t

n×p-matrix (n>p)

for t=1, . . . ,T

exploratory variables (clinical)

y

numeric vector of lengthn target variable (evolution of weight:

weight₃⁻weight₁ weight₁ )

(8)

Method presented in this talk

Features:

1 longitudinal dataintegration (withT small; hereT =3)

2 multidimensional data analysis withsimple graphical representations (similar to PCA)

3 designed tohighlight differences in the numeric target variable

4 designed tohighlight differences/commonalities between variable structure (rather than individual structure) between the different time steps

(9)

Method presented in this talk

Features:

(10)

Method presented in this talk

Features:

(11)

Method presented in this talk

Features:

(12)

Sommaire

3 Application

(13)

Factor analysis methods for multi-tables data integration

Standard methodsto analyze data such as(X_..t)_t=1,...,T:

Multiple Factor Analysis(MFA) [Escofier and Pagès, 2008]

: weights for tablesX_..t according to a measure of their variability (using first eigenvector of the PCA);

STATISandDUAL STATIS[Lavit et al., 1994,Abdi et al., 2012]

: larger importance to the most consensual tables⇒common representation which is thebest overall compromise. STATIS: compromise according to individuals andDUAL STATIS: compromise according to

correlations between variables.

(14)

Factor analysis methods for multi-tables data integration

Multiple Factor Analysis(MFA) [Escofier and Pagès, 2008]: weights for tablesX_..t according to a measure of their variability (using first eigenvector of the PCA);

STATISandDUAL STATIS[Lavit et al., 1994,Abdi et al., 2012]

: larger importance to the most consensual tables⇒common representation which is thebest overall compromise. STATIS: compromise according to individuals andDUAL STATIS: compromise according to

(15)

Factor analysis methods for multi-tables data integration

STATISandDUAL STATIS[Lavit et al., 1994,Abdi et al., 2012]: larger importance to the most consensual tables⇒common representation which is thebest overall compromise.

STATIS: compromise according to individuals andDUAL STATIS: compromise according to

(16)

Factor analysis methods for multi-tables data integration

STATISandDUAL STATIS[Lavit et al., 1994,Abdi et al., 2012]: larger importance to the most consensual tables⇒common representation which is thebest overall compromise. STATIS: compromise according to individuals andDUAL STATIS: compromise according to

(17)

Overall description of DUAL STATIS

Data: X=





 X_..1

...

X_..T







same number of variables, potentially different number of individuals data are supposed centered and

scaled for allt

inter-structure analysis analysis ofeC(T ×T similarity

matrix)

table weights (α₁, . . . , α_T)

compromise matrix Γ^c =P_T

t=1α_tΓ_t in whichΓ_t is the covariance matrix ofX..t

eig. dec. GSVD

compromise analysis

representations of variables and of individuals compromise positions or time-dependant positions

(18)

Overall description of DUAL STATIS

Data: X=





 X_..1

...

X_..T







scaled for allt

inter-structure analysis analysis ofeC(T×T similarity

matrix)

compromise matrix Γ^c =P_T

eig. dec. GSVD

compromise analysis

(19)

Overall description of DUAL STATIS

Data: X=





 X_..1

...

X_..T







scaled for allt

matrix)

compromise matrix Γ^c =PT

eig. dec. GSVD

compromise analysis

(20)

Overall description of DUAL STATIS

Data: X=





 X_..1

...

X_..T







scaled for allt

matrix)

compromise matrix Γ^c =PT

eig. dec.

GSVD

compromise analysis

(21)

Inter-structure analysis

Notations: ∀t =1, . . . , T,Γ_t = ¹_nX^>_..tX_..t andeΓ_t = _kΓ^Γ^t

tk_F

Similarity between time steps:e_C= (c_tt⁰)_t,t⁰_=1,...,T with c_tt⁰ =heΓ_t,eΓ_t⁰i_F

(cosinus betweenΓt andΓt⁰). Inter-structure analysis: Solve

u₁ = arg max

u=(u1,...,uT)^>∈R^T,kuk=1

XT

t=1

u_teΓt

2

F

Solution: u₁is the first eigenvector ofeC⇒compromise weights:

∀t =1, . . . ,T,α_t = ^P^u^1t

t0u_1t⁰

Γ^c =PT t=1α_teΓ_t

(22)

Inter-structure analysis

tk_F

(cosinus betweenΓt andΓt⁰).

Inter-structure analysis: Solve

u₁ = arg max

u=(u1,...,uT)^>∈R^T,kuk=1

XT

t=1

u_teΓt

2

F

∀t =1, . . . ,T,α_t = ^P^u^1t

t0u_1t⁰

(23)

Inter-structure analysis

tk_F

u₁ = arg max

u=(u1,...,uT)^>∈R^T,kuk=1

XT

t=1

u_teΓt

2

F

∀t =1, . . . ,T,α_t = ^P^u^1t

t0u_1t⁰

(24)

Inter-structure analysis

tk_F

u₁ = arg max

u=(u1,...,uT)^>∈R^T,kuk=1

XT

t=1

u_teΓt

2

F

∀t =1, . . . ,T,α_t = ^P^u^1t

t0u_1t⁰

(25)

Compromise analysis

GPCAof(eX,Ip,D)witheX_..t = ^√^X^..t

kΓtk_F andD= ¹_nDiag(α1, . . . , αT)⊗In

⇔eigendecomposition ofΓ^c Solutionwrites:

eX=PΛQ^> withP^>DP=Q^>Q=Ir andΛ =Diag(p

λ_k)_k=1,...,r

Representationsof:

observations: time-specific positionson thek-th principal component: k-th column ofF=PΛ =





 F₁

... F_T







compromise positionson thek-th principal component:k-th column ofF^c =P_T

t=1α_tF_..t

variables:compromise positions on circle of correlationsfor variablej onk-th axis:

√λk

σˆj Q_jk (time-specific positionscan also be derived)

(26)

Compromise analysis

⇔eigendecomposition ofΓ^c

Solutionwrites:

λ_k)_k=1,...,r

Representationsof:





 F₁

... F_T







t=1α_tF_..t

√λk

(27)

Compromise analysis

λ_k)_k=1,...,r

Representationsof:





 F₁

... F_T







t=1α_tF_..t

√λk

(28)

Compromise analysis

λ_k)_k=1,...,r

Representationsof:

observations: time-specific positionson thek-th principal component:

k-th column ofF=PΛ =





 F₁

...

F_T







t=1α_tF_..t

√λk

(29)

Compromise analysis

λ_k)_k=1,...,r

Representationsof:

observations: time-specific positionson thek-th principal component:

k-th column ofF=PΛ =





 F₁

...

F_T







t=1α_tF_..t

√λk

ˆσ_j Q_jk (time-specific positionscan also be derived)

(30)

SIR Regression framework

[Li, 1991]

Y =f(X^>a₁, . . . , X^>a_d, )

ford<pandf : R^d+1→R, an arbitrary (non linear) function.

S_Y|X =Span{a₁, . . . , a_d}is:Effective Dimension Reduction (EDR) space.

Equivalence between SIR and eigendecomposition

S_Y|Xis included in the space spanned by the firstdΓ-orthogonal eigenvectors of theΓ^e, withΓ =E

h(X−E(X))^TXi and Γ^e =E

E(Z|Y)^TE(Z|Y)

forZ = Γ^−1/2(X−E(X)).

(31)

SIR Regression framework

[Li, 1991]

Y =f(X^>a₁, . . . , X^>a_d, )

ford<pandf : R^d+1→R, an arbitrary (non linear) function.

S_Y|X =Span{a₁, . . . , a_d}is:Effective Dimension Reduction (EDR) space.

Equivalence between SIR and eigendecomposition

S_Y|Xis included in the space spanned by the firstdΓ-orthogonal eigenvectors of theΓ^e, withΓ =E

h(X−E(X))^TXi and Γ^e =E

E(Z|Y)^TE(Z|Y)

forZ = Γ^−1/2(X−E(X)).

(32)

SIR in practice

Estimation(whenn>p) compute¯x= ¹_nP_n

i=1x_iandΓ = ¹_nX^T(X−¯x)

split the range ofY intoHdifferent slices: τ₁, ...τ_H and estimate

G=





 1 n_h

X

i:yi∈τh

z_i







h=1,...,H

withn_h =|{i : y_i ∈τ_h}|and

Γ^e =G^>MG withM=Diag_n

1

n, . . . ,ⁿ_n^H

generalized eigendecompositionofΓ^e with normΓ

(33)

SIR in practice

G=





 1 n_h

X

i:yi∈τh

z_i







h=1,...,H

1

n, . . . ,ⁿ_n^H

(34)

SIR in practice

G=





 1 n_h

X

i:yi∈τh

z_i







h=1,...,H

1

n, . . . ,ⁿ_n^H

(35)

Back to our problem: multiway SIR

Basic ideas:

perform DUAL STATIS analysis on center of gravity of the slices (G_..t) instead of the original variables

compromise analysis is similar to finding a compromise EDR space using slices make the method similar to FDA but other estimates ofΓ^e_t (not based on slices) could be used

(36)

Overview of multiway SIR

Data:eX=





 X_..1

...

X_..T





 and y= (y₁, . . . , y_n)^>

same number of variables, potentially different number of individuals

inter-structure analysis analysis ofeC(T ×T similarity

matrixbetween the covariance matrix ofG..t,Γ^e_t)

compromise matrix Γ^e,c=PT

t=1α_tΓ^e_t eig. dec.

GSVD

compromise analysis

(37)

Overview of multiway SIR

Data:eX=





 X_..1

...

X_..T





 and y= (y₁, . . . , y_n)^>

t=1α_tΓ^e_t eig. dec.

GSVD

compromise analysis

(38)

Overview of multiway SIR

Data:eX=





 X_..1

...

X_..T





 and y= (y₁, . . . , y_n)^>

t=1αtΓ^e_t

eig. dec. GSVD

compromise analysis

(39)

Overview of multiway SIR

Data:eX=





 X_..1

...

X_..T





 and y= (y₁, . . . , y_n)^>

t=1αtΓ^e_t eig. dec.

GSVD

compromise analysis

(40)

Inter-structure analysis

Notations:∀t =1, . . . , T,Γt = ¹_nX^>_..t(X_..t−1_nx^>_t ),Z_..t =

X_..t −1_nx^>_t Γ^−1/2_t , G_..t =₁

nh

P

i:yi∈τhz_i

h=1,...,H andΓ^e_t =G^>_..tM G_..t andeΓ^e_t = ^Γ

e t

kΓ^e_tk_F.

Similarity between time steps:eC= (c_tt⁰)_t,t⁰_=1,...,T with c_tt⁰ =heΓ^e_t,eΓ^e_t0i_F

(cosinus betweenΓ^e_t andΓ^e_t0). Inter-structure analysis: Solve

u₁= arg max

u=(u1,...,uT)^>∈R^T,kuk=1

T

X

t=1

u_teΓ^e_t

2

F

∀t =1, . . . ,T,α_t = ^P^u^1t

t0u_1t⁰

Γ^e,c =PT t=1α_teΓ^e_t

(41)

Inter-structure analysis

X_..t −1_nx^>_t Γ^−1/2_t , G_..t =₁

nh

P

i:yi∈τhz_i

e t

kΓ^e_tk_F. Similarity between time steps:eC= (c_tt⁰)_t,t⁰_=1,...,T with

c_tt⁰ =heΓê_t,eΓê_t0i_F (cosinus betweenΓê_t andΓê_t0).

u₁= arg max

u=(u1,...,uT)^>∈R^T,kuk=1

T

X

t=1

u_teΓ^e_t

2

F

∀t =1, . . . ,T,α_t = ^P^u^1t

t0u_1t⁰

(42)

Inter-structure analysis

X_..t −1_nx^>_t Γ^−1/2_t , G_..t =₁

nh

P

i:yi∈τhz_i

e t

u₁= arg max

u=(u1,...,uT)^>∈R^T,kuk=1

T

X

t=1

u_teΓ^e_t

2

F

∀t =1, . . . ,T,α_t = ^P^u^1t

t0u_1t⁰

(43)

Inter-structure analysis

X_..t −1_nx^>_t Γ^−1/2_t , G_..t =₁

nh

P

i:yi∈τhz_i

e t

u₁= arg max

u=(u1,...,uT)^>∈R^T,kuk=1

T

X

t=1

u_teΓ^e_t

2

F

∀t =1, . . . ,T,α_t = ^P^u^1t

t0u_1t⁰

Γ^e,c=P_T

t=1α_teΓ^e_t

(44)

Compromise analysis

GPCAof(Ge,Ip,D)withGe_..t = √^G^..t

kΓ^e_tk_F andD=Diag(α1, . . . , αT)⊗M

⇔eigendecomposition ofΓ^e,c Solutionwrites:

Ge=PΛQ^> withP^>DP=Q^>Q=Ir andΛ =Diag(p

λ_k)_k=1,...,r

Representationsof:

slices: time-specific positionson thek-th principal component: k-th column ofF=PΛ =





 F₁

... F_T







t=1α_tF_..t

√λ_k

(45)

Compromise analysis

⇔eigendecomposition ofΓ^e,c

Solutionwrites:

λ_k)_k=1,...,r

Representationsof:





 F₁

... F_T







t=1α_tF_..t

√λ_k

(46)

Compromise analysis

λ_k)_k=1,...,r

Representationsof:





 F₁

... F_T







t=1α_tF_..t

√λ_k

(47)

Compromise analysis

λ_k)_k=1,...,r

Representationsof:





 F₁

...

F_T







compromise positionson thek-th principal component:k-th column ofF^c =PT

t=1α_tF_..t

√λ_k

(48)

Compromise analysis

λ_k)_k=1,...,r

Representationsof:





 F₁

...

F_T







compromise positionson thek-th principal component:k-th column ofF^c =PT

t=1α_tF_..t

(49)

Sommaire

3 Application

(50)

Application of multiway SIR on Diogenes dataset I

H =5,interstructure analysis:

(51)

Application of multiway SIR on Diogenes dataset II

H =5,compromise analysis: number of components

2 components are enough but we will analyze 4 for a deeper understanding of the data.

(52)

Application of multiway SIR on Diogenes dataset III

H =5,compromise analysis: analysis of the slices (compromise)

(53)

Application of multiway SIR on Diogenes dataset IV

H =5,compromise analysis: analysis of the variable (compromise)

(54)

Application of multiway SIR on Diogenes dataset V

H =5,compromise analysis: longitudinal analysis

(55)

Conclusion

We have presented an exploratory analysis method based on DUAL STATIS

able to focus on a numeric variable of interest similarly to SIR

able to explain longitudinal evolutions when the number of time steps is small

Future work

an R package is under development

only valid forn≥p: regularization approach is under study to allows for the analysis ofn<pcases

(56)

Abdi, H., Williams, L., Valentin, D., and Bennani-Dosse, M. (2012).

STATIS and DISTATIS: optimum multitable principal component analysis and three way metric multidimensional scaling.

Wiley Interdisciplinary Reviews: Computational Statistics, 4(2):124–167.

Escofier, B. and Pagès, J. (2008).

Analyses Factorielles Simples et Multiples.

Dunod.

Lavit, C., Escoufier, Y., Sabatier, R., and Traissac, P. (1994).

The ACT (STATIS method).

Computational Statistics and Data Analysis, 18(1):97–119.

Li, K. (1991).

Sliced inverse regression for dimension reduction.

Journal of the American Statistical Association, 86(414):316–342.