Arthur Tenenhaus

(CentraleSupelec, Laboratoire des Signaux et Systèmes, France)

https://s3-seminar.github.io/seminars/arthur-tenenhaus

Title — Structured data analysis

Abstract — In contrast to standard data that is structured by a single individuals variables data matrix, structured data are characterized by multiple and heterogeneous sources of information, interconnected, potentially of high dimensions. In addition, each source of information may also have a complex structure (e.g. tensor structure). The need to analyze the data by taking into account their natural structure appears to be essential but requires the development of new statistical techniques that constitute the core of my current research for many years. More specifically, I will present a unified framework for multiblock, multigroup and multiway data analysis through Regularized Generalized Canonical Correlation Analysis.

Transcript

1. Structured data analysis with RGCCA
   Arthur Tenenhaus 2015/02/13
   

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2. Overview of the presentation
   X1
   X2
   XJ
   n
   p2
   p3
   p1
   ...
   Part I. Multi-block analysis
   X1
   X2
   XJ
   n
   p2
   p3
   p1
   ...
   Part II. Multi-block and
   Multi-way analysis
   X2
   X21
   X1
   X2
   XI
   n1
   p
   ...
   Part III. Multi-group analysis
   n2
   nI
   

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3. X1
   X2
   XJ
   n
   p2
   p3
   p1
   ...
   Part I. Multi-block analysis
   M. Tenenhaus
   (HEC, Paris)
   V. Guillemot
   (ICM)
   T. Löfstedt
   (CEA, Neurospin)
   C. Philippe
   (IGR)
   V. Frouin
   (CEA, Neurospin)
   P. Groenen
   (Erasmus University,
   Rotterdam)
   

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4. 4
   Economic inequality and political instability Data
   from Russett (1964)
   Agricultural inequality
   GINI : Inequality of land
   distributions
   FARM : % farmers that own half
   of the land (> 50)
   RENT : % farmers that rent all
   their land
   Industrial development
   GNPR : Gross national product
   per capita ($ 1955)
   LABO : % of labor force
   employed in agriculture
   INST : Instability of executive
   (45-61)
   ECKS : Nb of violent internal war
   incidents (46-61)
   DEAT : Nb of people killed as a
   result of civic group
   violence (50-62)
   D-STAB : Stable democracy
   D-UNST : Unstable democracy
   DICT : Dictatorship
   Economic inequality Political instability
   

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5. Economic inequality and political instability
   (Data from Russett, 1964)
   Gini Farm Rent Gnpr Labo Inst Ecks Deat Demo
   Argentine 86.3 98.2 32.9 374 25 13.6 57 217 2
   Australie 92.9 99.6 * 1215 14 11.3 0 0 1
   Autriche 74.0 97.4 10.7 532 32 12.8 4 0 2
   
   France 58.3 86.1 26.0 1046 26 16.3 46 1 2
   
   Yougoslavie 43.7 79.8 0.0 297 67 0.0 9 0 3
   X1
   X2
   X3
   5
   1 = Stable democracy
   2 = Unstable democracy
   3 = Dictatorship
   Three data blocks
   

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6. GINI
   FARM
   RENT
   GNPR
   LABO
   Agricultural inequality (X1
   )
   Industrial development (X2
   )
   ECKS
   DEAT
   D-STB
   D-INS
   INST
   DICT
   Political instability (X3
   )
   Agr.
   ineq.
   Ind.
   dev.
   Pol.
   inst.
   C13
   = 1
   C23
   = 1
   C12
   = 0
   Path diagram
   6
   

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7. Block components
   1
   = 1
   1
   = 11
   + 12
   + 13
   
   2
   = 2
   2
   = 21
   + 22
   
   3
   = 3
   3
   = 31
   + 32
   + 33
   +
   34
   . + 35
   . + 36
   
   Block components should verified two properties at the same
   time:
   (i) Block components explain well their own block.
   (ii) Block components are as correlated as possible for
   connected blocks.
   

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8. Covariance-based criteria
   cjk
   = 1 if blocks are linked, 0 otherwise and cjj
   = 0
   maximize
   ,
   
   cor(
   
   ,
   
   )
   maximize
   ,
   
   cor2(
   
   ,
   
   )
   maximize
   ,
   
   |cor
   
   ,
   
   |
   maximize
   all ‖‖=1
   ,
   
   cov(
   
   ,
   
   )
   maximize
   all ‖‖=1
   ,
   
   cov2(
   
   ,
   
   )
   maximize
   all ‖‖=1
   ,
   
   |cov
   
   ,
   
   |
   SUMCOR (Horst, 1961)
   SSQCOR (Mathes, 1993 ; Hanafi, 2004)
   SABSCOR (Mathes, 1993 ; Hanafi, 2004)
   SUMCOV (Van de Geer, 1984)
   SSQCOV (Hanafi & Kiers, 2006)
   SABSCOV (Krämer, 2006)
   Some modified multi-block methods
   GENERALIZED CANONICAL CORRELATION ANALYSIS
   GENERALIZED CANONICAL COVARIANCE ANALYSIS
   cov2
   
   ,
   
   = var
   
   cor2(
   
   ,
   
   )var
   
   Some multi-block methods
   

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9. Covariance-based criteria
   cjk
   = 1 if blocks are linked, 0 otherwise and cjj
   = 0
   maximize
   all var =1
   ,
   
   cov(
   
   ,
   
   )
   maximize
   all var =1
   ,
   
   cov2(
   
   ,
   
   )
   maximize
   all var =1
   ,
   
   |cov
   
   ,
   
   |
   maximize
   all ‖‖=1
   ,
   
   cov(
   
   ,
   
   )
   maximize
   all ‖‖=1
   ,
   
   cov2(
   
   ,
   
   )
   maximize
   all ‖‖=1
   ,
   
   |cov
   
   ,
   
   |
   SUMCOR:
   SSQCOR:
   SABSCOR:
   SUMCOV:
   SSQCOV:
   SABSCOV:
   cov2
   
   ,
   
   = var
   
   cor2(
   
   ,
   
   )var
   
   

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10. RGCCA optimization problem
    argmax
    1,2,…,
    
    g cov
    
    ,
    
    
    ≠
    1 −
    var
    
    +
    
    2
    = 1, = 1, … ,
    Subject to the constraints
    and:
    identity (Horst sheme)
    square (Factorial scheme)
    abolute value (Centroid scheme)
    g
    
    
     
    
    
    Shrinkage constant between 0 and 1
    j
     
    
    
    
    
    otherwise
    0
    connected
    is
    and
    if
    1
    k
    j
    X
    X
    jk
    c
    where: A monotone convergent algorithm
    Schäfer and Strimmer formula can be used for an
    optimal determination of the shrinkage constants
    • Tenenhaus A. and Tenenhaus M., Regularized Generalized Canonical Correlation Analysis, Psychometrika, vol. 76, Issue 2, pp. 257-284, 2011
    • Tenenhaus A., Philippe C., Frouin V., Kernel Generalized Canonical Correlation Analysis, Computational Statistics and Data Analysis, in revision.
    • Tenenhaus A. and Guillemot V. (2013): RGCCA Package. http://cran.project.org/web/packages/RGCCA/index.html
    

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11. Method Criterion Constraints
    PLS regression 1 1 2 2
    Maximize Cov( , )
    X a X a
    1 2
    1
     
    a a
    Canonical
    Correlation
    Analysis
    1 1 2 2
    Maximize Cor( , )
    X a X a
    1 1 2 2
    Var( ) Var( ) 1
     
    X a X a
    Redundancy
    analysis of X1
    with
    respect to X2
    1/2
    1 1 2 2 1 1
    Maximize
    Cor( , )Var( )
    X a X a X a
    1
    2 2
    1
    Var( ) 1
    
    
    a
    X a
    Special cases
    Components X1
    a1
    and X2
    a2
    are well correlated.
    No stability condition for
    2nd component
    1st component is stable
    argmax
    1,2
    cov(1
    1
    , 2
    2
    )
    1 −
    var
    
    +
    
    2
    = 1, = 1,2
    Subject to the constraints
    Choice of the shrinkage constant j
    (part 1)
    

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12. Choice of the shrinkage constant j
    (part 2)
    0 1
    Favoring
    correlation
    Favoring
    stability
    j
    Schäfer and Strimmer formula can be used for an
    optimal determination of the shrinkage constants
    argmax
    1,2,…,
    
    g cov
    
    ,
    
    
    ≠
    1 −
    var
    
    +
    
    2
    = 1, = 1, … ,
    Subject to the constraints
    

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13. Choice of the design matrix C
    Hierarchical models
    (a) One second order block (b) Several second order blocks
    max
    1,2,…,
    ≠
    
    g cov
    
    , +1
    +1
    1 −
    var
    
    +
    
    2
    = 1, = 1, … , + 1
    max
    1,2,…,
    =1
    1
    =1+1
    
    
    g cov
    
    ,
    
    1 −
    var
    
    +
    
    2
    = 1, = 1, … ,
    1, …, J1
    = Predictor blocks
    Very often:
    J1+1, …, J = Response Blocks
    

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14. PLS Regression Wold S., Martens & Wold H. (1983): The multivariate calibration problem in chemistry solved by the PLS method. In Proc.
    Conf. Matrix Pencils, Ruhe A. & Kåstrøm B. (Eds), March 1982, Lecture Notes in Mathematics, Springer Verlag,
    Heidelberg, p. 286-293.
    Redundancy analysis Barker M. & Rayens W. (2003): Partial least squares for discrimination, Journal of Chemometrics, 17, 166-173.
    Regularized CCA Vinod H. D. (1976): Canonical ridge and econometrics of joint production. Journal of Econometrics, 4, 147–166.
    Inter-battery factor analysis Tucker L.R. (1958): An inter-battery method of factor analysis, Psychometrika, vol. 23, n°2, pp. 111-136.
    MCOA Chessel D. and Hanafi M. (1996): Analyse de la co-inertie de K nuages de points. Revue de Statistique Appliquée, 44, 35-60
    SSQCOV Hanafi M. & Kiers H.A.L. (2006): Analysis of K sets of data, with differential emphasis on agreement between and within
    sets, Computational Statistics & Data Analysis, 51, 1491-1508.
    SUMCOR Horst P. (1961): Relations among m sets of variables, Psychometrika, vol. 26, pp. 126-149.
    SSQCOR Kettenring J.R. (1971): Canonical analysis of several sets of variables, Biometrika, 58, 433-451
    MAXDIFF Van de Geer J. P. (1984): Linear relations among k sets of variables. Psychometrika, 49, 70-94.
    PLS path modeling Tenenhaus M., Esposito Vinzi V., Chatelin Y.-M., Lauro C. (2005): PLS path modeling. Computational Statistics and Data
    (mode B) Analysis, 48, 159-205.
    Generalized Orthogonal Vivien M. & Sabatier R. (2003): Generalized orthogonal multiple co-inertia analysis (-PLS): new multiblock component
    MCOA and regression methods, Journal of Chemometrics, 17, 287-301.
    Caroll’s GCCA Carroll, J.D. (1968): A generalization of canonical correlation analysis to three or more sets of variables, Proc. 76th
    Conv. Am. Psych. Assoc., pp. 227-228.
    special cases of RGCCA (among others)
    two-block case
    multi-block case
    

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15. monotone convergent algorithms for these criteria
    argmax
    1,2,…,
    
    g cov
    
    ,
    
    
    ≠
    1 −
    var
    
    +
    
    2
    = 1, = 1, … ,
    Subject to
    Two key ingredients: (i) Block relaxation
    (ii) Majorization by Minorization
    

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16. The RGCCA algorithm (primal version)
    j
    j
    j
    a
    X
    y 
    Outer Estimation
    (explains the block)
        1
    var
    1 2
    
    
    
    j
    j
    j
    j
    j
    a
    a
    X 
    
    Initial
    step j
    a
     
     
    j
    t
    j
    j
    j
    t
    j
    j
    j
    t
    j
    j
    t
    j
    j
    j
    t
    j
    j
    j
    z
    X
    I
    X
    X
    n
    X
    z
    z
    X
    I
    X
    X
    n
    a
    1
    1
    1
    1
    1
    1
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    Iterate until
    convergence
    of the criterion
    cjk
    = 1 if blocks are linked, 0 otherwise and cjj
    = 0
    Inner
    Estimation
    (explains
    relation
    between
    block)
    k
    j
    k
    jk
    j
    e y
    z 
    
    
    Choice of weights ejk
    :
    - Horst :
    - Centroid :
    - Factorial :
    jk
    jk
    c
    e 
     
     
    k
    j
    jk
    jk
    c
    e y
    y ,
    cor
    sign
    
     
    k
    j
    jk
    jk
    c
    e y
    y ,
    cov
    
    Dimension =
    ×
    

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17. The RGCCA algorithm (dual version)
    Initial
    step j
    α
     
     
    j
    j
    t
    j
    j
    j
    t
    j
    j
    t
    j
    j
    j
    t
    j
    j
    j
    j
    z
    I
    X
    X
    n
    X
    X
    z
    z
    I
    X
    X
    n
    α
    1
    1
    1
    1
    1
    1
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    Iterate until
    convergence
    of the criterion
    cjk
    = 1 if blocks are linked, 0 otherwise and cjj
    = 0
    Inner
    Estimation
    (explains
    relation
    between
    block)
    k
    j
    k
    jk
    j
    e y
    z 
    
    
    Choice of weights ejk
    :
    - Horst :
    - Centroid :
    - Factorial :
    jk
    jk
    c
    e 
     
     
    k
    j
    jk
    jk
    c
    e y
    y ,
    cor
    sign
    
     
    k
    j
    jk
    jk
    c
    e y
    y ,
    cov
    
    j
    t
    j
    j
    j
    α
    X
    X
    y 
    Outer Estimation
    (explains the block)
     
      1
    )
    1
    ( 1 
    
    
    j
    t
    j
    j
    n
    j
    j
    t
    j
    j
    t
    j
    α
    X
    X
    I
    X
    X
    α 
    
    Dimension = ×
    j
    t
    j
    j
    α
    X
    a 
    

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18. GINI
    FARM
    RENT
    GNPR
    LABO
    Agricultural inequality (X1
    )
    Industrial development (X2
    )
    ECKS
    DEAT
    D-STB
    INST
    DICT
    Political instability (X3
    )
    Agr.
    ineq.
    Ind.
    dev.
    Pol.
    inst.
    Weight vectors
    18
    0.66
    0.74
    0.10
    0.69
    -0.72
    0.17
    0.44
    0.48
    -0.56
    0.49
    Corr = 0.428
    Corr = -0.767
    small dimensional block settings ⟹ primal algorithm for RGCCA
    

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19. Bootstrap confidence intervals
    19
    GINI
    FARM
    RENT
    GNPR
    LABO
    ECKS
    DEAT
    D-STB
    INST
    DICT
    Agr.
    ineq.
    Ind.
    dev.
    Pol.
    inst.
    0.66
    0.74
    0.10
    0.69
    -0.72
    0.17
    0.44
    0.48
    -0.56
    0.49
    Corr = 0.428
    Corr = -0.767
    

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20. Data vizualization
    20
    Agricultural inequality
    Industrial development
    These countries
    have known a
    period of
    dictatorship
    after 1964.
    Greece : colonels’ dictatorship 1967-1974
    Chili : Pinochet's military regime 1973-1990
    Argentine : military dictatorship 1976-1983
    Brasil : Branco’s military dictactorship 1964-1985
    

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21. 21
    Glioma Cancer Data
    (Department of Pediatric Oncology of the Gustave Roussy Institute)
    Gene 1 Gene 2 … Gene 15201 CGH1 … CGH 1909 Localization
    Patient 1 0.18 -0.21 -0.73 0.00 -0.55 Hemisphere
    Patient 2 1.15 -0.45 0.27 -0.30 0.00 Midline
    Patient 3 1.35 0.17 0.22 0.33 0.64 DIPG
    
    
    Patient 53 1.39 0.18 … -0.17 0.00 … 0.43 Hemisphere
    Transcriptomic data (X1
    )
    CGH data (X2
    )
    outcome (X3
    )
    

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22. Glioma Cancer Data: from an RGCCA viewpoint
    (Department of Pediatric Oncology of the Gustave Roussy Institute)
    High dimensional block settings ⟹ dual algorithm for RGCCA
    Gene 1 … Gene 15201
    Patient 1 0.18 -0.73
    Patient 2 1.15 0.27
    Patient 3 1.35 0.22
    
    Patient 53 1.39 -0.17
    CGH1 … CGH 1909
    Patient 1 0.00 -0.55
    Patient 2 -0.30 0.00
    Patient 3 0.33 0.64
    
    Patient 53 0.00 0.43
    2
    1
    3
    Hemisphere DIPG
    Patient 1 1 0
    Patient 2 0 0
    Patient 3 0 1
    
    Patient 53 1 0
    RGCCA with factorial scheme - 1
    = 1, 2
    = 1 and 3
    = 0
    C13
    = 1
    C23
    = 1
    C12
    = 1
    C12
    = 0
    

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23. View Slide

24. Block components
    1
    =
    
    = 11
    
    
    + ⋯ + 1,15201
    
    
    2
    =
    
    = 21
    
    
    + ⋯ + 2,1909
    
    
    
    =
    
    = 31
    + 32
    
    Block components should verified three properties at the same
    time:
    (i) Block components well explain their own block.
    (ii) Block components are as correlated as possible for
    connected blocks.
    (iii) Block components are built from sparse
    

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25. Behavioral data
    (Clinic, psychometric)
    Intermediate phenotype
    Final phenotype
    Genotype
    Functional MRI
    Gene Expression
    Structured variable selection for RGCCA
    

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26. (Structured) variable selection for RGCCA
    argmax
    1,2,…,
    
    g cov
    
    ,
    
    
    ≠
    subject to
    
    
    
    = 1, = 1, … ,
    Ω(
    ) ≤
    , = 1, … ,
    • LASSO: Ω
    = 1
    Ω
    =
    =1
    
    
    +
    =1
    
    
    − ,−1
    Ω
    =
    ∈
    ag 2
    • Group LASSO:
    • Fused LASSO:
    • Tenenhaus A., Philippe C., Guillemot V., Lê Cao K.-A., Grill J., Frouin V., Variable Selection for Generalized Canonical Correlation Analysis, Biostatistics, 15 (3),
    569-583, 2014.
    • Löfstedt T., Hadj-Salem F., Guillemot V., Philippe C., Duchesnay E., Frouin V., and Tenenhaus A., (2014). Structured variable selection for generalized
    canonical correlation analysis. In: Proceedings of the 8th International Conference on Partial Least Squares and Related Methods (PLS14), Paris, France.
    • Tenenhaus A. and Guillemot V. (2013): RGCCA Package. http://cran.project.org/web/packages/RGCCA/index.html
    

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27. Signature stability
    

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28. 28
    Predictive performances
    

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29. Visualization
    GE1
    CGH1
    

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30. X1
    X2
    XJ
    n
    p2
    p3
    p1
    ...
    Part II. Multi-block and
    Multi-way analysis
    X2
    X21
    G. Lechuga
    (CentraleSupélec, L2S)
    L. Le Brusquet
    (CentraleSupélec, L2S)
    L. Puybasset, V. Perlbarg & D. Galanaud
    Hôpital La Pitié-Salpêtrière
    

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31. 31
    Multimodal neuroimaging from a multiblock viewpoint
    (Brain and Spine Institute, La pitié Salpêtrière Hospital)
    2
    1
    3
    C13
    = 1
    C23
    = 1
    C12
    = 1
    C13
    = 0
    Anatomical MRI (X1
    )
    Functional MRI (X2
    )
    Behavior (X3
    )
    Anat 1 … Anat p1
    Patient 1 0.18 -0.73
    Patient 2 1.15 0.27
    Patient 3 1.35 0.22
    
    Patient n 1.39 -0.17
    fMRI 1 … fMRI p1
    Patient 1 0.00 -0.55
    Patient 2 -0.30 0.00
    Patient 3 0.33 0.64
    
    Patient n 0.00 0.43
    BEH 1 … BEH p3
    Patient 1 0.18 -0.73
    Patient 2 1.15 0.27
    Patient 3 1.35 0.22
    
    Patient n 1.39 -0.17
    p1
    ~104
    p2
    ~104
    p3
    ~10
    n ~100
    n ~100
    

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32. X5
    Final Phenotype
    X1
    X2
    X3
    X4
    Anatomical MRI Diffusion MRI Functional MRI
    PET
    p1
    p1
    p1
    p1
    p2
    From Multiblock data to …
    Multiway RGCCA (MGCCA)
    RGCCA algorithm
    a1
    a2
    a3
    a5
    a4
    4×p1
    parameters to estimate
    n
    • Tenenhaus A., Le Brusquet L. Regularized Generalized Canonical Correlation Analysis extended to three way data, International Conference of the ERCIM WG on
    Computational and Methodological Statistics, 2014
    • Tenenhaus A., Le Brusquet L. Three-way Regularized Generalized Canonical Correlation Analysis, ThRee-way methods In Chemistry And Psychology, (TRICAP) ,2015
    

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33. X5
    … to Multiblock / Multiway data
    X1 2
    1
    C12
    = 1
    J
    1
    b
    K
    1
    b
    2
    b
    X5
    X1
    X2
    X3
    X4
    P1
    = P2
    =
    and
    
    
    = 1, = 1,2
    1
    = 1
    ⊗ 1
    
    s.c.
    max
    1, 2
    cov(1
    1
    , 2
    2
    )
    

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34. P2
    … to Multiblock / Multiway data
    P1 2
    1
    C12
    = 1
    J
    1
    b
    K
    1
    b
    2
    b
    MGCCA
    b2
    K
    1
    b J
    1
    b
    4 + p1
    parameters to estimate
    instead of 4 × p1
    K J
    1 1 1
    b b b
     
    

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35. MGCCA optimization problem
    
    
    = 1, = 1, … ,
    
    =
    ⊗
    
    , = 1, … ,
    s.c.
    max
    1,…,
    
    g cov(
    
    ,
    
    )
    
    ≠
    1
    ..1
    P
    1
    ..k
    P
    1
    1
    ..K
    P
    3
    ..1
    P
    3
    ..k
    P
    3
    3
    ..K
    P
    2
    ..1
    P
    2
    ..k
    P
    2
    2
    ..K
    P
    1
    3
    2
    C13
    = 1
    C23
    = 1
    C12
    = 1
    C13
    = 0
    J
    1
    b
    K
    1
    b
    J
    2
    b
    K
    2
    b
    J
    3
    b
    K
    3
    b
    1
    2
    3
    

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36. MGCCA algorithm
    y P b
    
    j j j
    Outer component
    (summarizes the block)
    Initial
    step j
    b
    Iterate until
    convergence
    of the criterion
    cjk
    = 1 if blocks/groups are connected and 0 otherwise
    Inner
    component
    (take into
    account
    relationships
    between
    blocks)
    j jk k
    k j
    z e y
    
     
     
    1
    , arg max cov( , )
    b b b
    b
    b b P b z
     
    
    
    K J
    K J
    j j j j
    Choice of weights ejk
    :
    - Horst :
    - Centroid :
    - Factorial :
    jk jk
    e =c
     
    sign cov( , )
    ik jk j k
    e c y y
    
    cov( , )
    jk jk j k
    e c y y
    
    ...,
    j
    j j
    j ..1 ..K
    ,
    P P P
     
      
    
    = 1 and
    =
    ⊗
    
    
     
    ,
    K J
    j j
    b b are obtained by SVD
    

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37. MGCCA results
    P2
    P1 2
    1
    J
    1
    b
    K
    1
    b
    2
    b
    Predict the long term recovery of patients after traumatic brain injury
    Influence of spatial positions Influence of the
    modalities
    J
    1
    b
    K
    1
    b
    Discriminating
    voxels within
    the white matter
    bundles
    Bad prognosis
    Good prognosis
    Control
    

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38. X1
    X2
    XI
    n1
    p
    ...
    Part III. Multi-group analysis
    n2
    nI
    M. Tenenhaus
    (HEC, Paris)
    

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39. Part III: multigroup data analysis
    • SETTINGS: The same set of
    variables are measured on individuals
    structured in several groups. groups
    are centered and normalized (unit
    norm)
    • OBJECTIVE: investigate the
    relationships between variables
    within the various groups.
    X2
    n
    1
    p
    X2
    n
    2
    n
    I
    X2
    X2
    Tenenhaus, A. and Tenenhaus, M. (2014). Regularized Generalized Canonical Correlation Analysis for multiblock or
    multigroup data analysis. European Journal of Operational Research, 238 :391–403.
    X2
    X2
    

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40. RGCCA for multiblock data analysis
    argmax
    1,2,…,
    
    g cov
    
    ,
    
    
    ≠
    1 −
    var
    
    +
    
    2
    = 1, = 1, … ,
    Subject to the constraints
    Block
    component
    cov2
    
    ,
    
    = var
    
    cor2(
    
    ,
    
    )var
    
    Block components should verified two properties at the same
    time:
    (i) Block components well explain their own block.
    (ii) Block components are as correlated as possible for
    connected blocks.
    

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41. RGCCA for multigroup data analysis
    argmax
    1,…,
    
    g
    
    
    ,
    
    
    
    ≠
    (1 −
    )‖
    
    ‖2 +
    ‖
    ‖2 = 1, = 1, … ,
    s.c.
    
    
    
    ,
    
    
    = cos
    
    
    ,
    
    
    ×
    
    
    × ‖
    
    
    ‖
    Group loadings and group components should verified the
    following properties at the same time:
    • Group component
    well explains their own block.
    • Small angle between loadings if groups are connected
    Similar Loadings
    Group
    loadings
    Group
    components
    

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42. Conclusion
    X1
    X2
    XJ
    n
    p2
    pJ
    p1
    ...
    Multiblock analysis
    X1
    X2
    XJ
    n
    p2
    p3
    p1
    ...
    Multiblock/Multiway analysis
    X2
    X21
    X1
    X2
    XI
    n1
    p
    ...
    Multigroup analysis
    n2
    nI
    RGCCA for multiblock,
    multigroup or multiway
    data allows analyzing
    the data in their natural
    (but complex) structure.
    

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