Skip to main content
Log in

Clusterwise analysis for multiblock component methods

  • Regular Article
  • Published:
Advances in Data Analysis and Classification Aims and scope Submit manuscript

Abstract

Multiblock component methods are applied to data sets for which several blocks of variables are measured on a same set of observations with the goal to analyze the relationships between these blocks of variables. In this article, we focus on multiblock component methods that integrate the information found in several blocks of explanatory variables in order to describe and explain one set of dependent variables. In the following, multiblock PLS and multiblock redundancy analysis are chosen, as particular cases of multiblock component methods when one set of variables is explained by a set of predictor variables that is organized into blocks. Because these multiblock techniques assume that the observations come from a homogeneous population they will provide suboptimal results when the observations actually come from different populations. A strategy to palliate this problem—presented in this article—is to use a technique such as clusterwise regression in order to identify homogeneous clusters of observations. This approach creates two new methods that provide clusters that have their own sets of regression coefficients. This combination of clustering and regression improves the overall quality of the prediction and facilitates the interpretation. In addition, the minimization of a well-defined criterion—by means of a sequential algorithm—ensures that the algorithm converges monotonously. Finally, the proposed method is distribution-free and can be used when the explanatory variables outnumber the observations within clusters. The proposed clusterwise multiblock methods are illustrated with of a simulation study and a (simulated) example from marketing.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  • Abdi H, Williams L (2012) Partial least squares methods: partial least squares correlation and partial least square regression. In: Reisfeld B, Mayeno A (eds) Methods in molecular biology: computational toxicology. Springer, New York, pp 549–579

    Google Scholar 

  • Bock H (1969) The equivalence of two extremal problems and its application to the iterative classification of multivariate data. In: Vortragsausarbeitung, Tagung. Mathematisches Forschungsinstitut Oberwolfach

  • Bougeard S, Cardinal M (2014) Multiblock modeling for complex preference study. Application to European preferences for smoked salmon. Food Qual Prefer 32:56–64

    Article  Google Scholar 

  • Bougeard S, Hanafi M, Qannari E (2007) ACPVI multibloc. Application à des données d’épidémiologie animale. Journal de la Société Française de Statistique 148:77–94

    Google Scholar 

  • Bougeard S, Qannari E, Lupo C, Hanafi M (2011a) From multiblock partial least squares to multiblock redundancy analysis. A continuum approach. Informatica 22:11–26

    MathSciNet  MATH  Google Scholar 

  • Bougeard S, Qannari E, Rose N (2011b) Multiblock redundancy analysis: interpretation tools and application in epidemiology. J Chemom 25:467–475

    Article  Google Scholar 

  • Bry X, Verron T, Redont P, Cazes P (2012) THEME-SEER: a multidimensional exploratory technique to analyze a structural model using an extended covariance criterion. J Chemom 26:158–169

    Article  Google Scholar 

  • Charles C (1977) Régression typologique et reconnaissance des formes. PhD thesis, University of Paris IX, France

  • De Roover K, Ceulemans C, Timmerman M (2012) Clusterwise simultaneous component analysis for analyzing structural differences in multivariate multiblock data. Psychol Methods 17:100–119

    Article  Google Scholar 

  • DeSarbo W, Cron W (1988) A maximum likelihood methodology for clusterwise linear regression. J Classif 5:249–282

    Article  MathSciNet  MATH  Google Scholar 

  • Diday E (1976) Classification et sélection de paramètres sous contraintes. Technical report, IRIA-LABORIA

  • Dolce P, Esposito Vinzi V, Lauro C (2016) Path directions incoherence in PLS path modeling: a prediction-oriented solution. In: Abdi H, Esposito Vinzi V, Russolillo G, Saporta G, Trinchera L (eds) The multiple facets of partial least squares and related methods. Springer proceedings in mathematics & statistics. Springer, Berlin, pp 59–59

    Google Scholar 

  • Hahn C, Johnson M, Hermann AFA (2002) Capturing customer heterogeneity using finite mixture PLS approach. Schmalenbach Bus Rev 54:243–269

    Article  Google Scholar 

  • Hubert H, Arabie P (1985) Comparing partitions. J Classif 2:193–218

    Article  MATH  Google Scholar 

  • Hwang H, Takane Y (2004) Generalized structured component analysis. Psychometrika 69:81–99

    Article  MathSciNet  MATH  Google Scholar 

  • Hwang H, DeSarbo S, Takane Y (2007) Fuzzy clusterwise generalized structured component analysis. Psychometrika 72:181–198

    Article  MathSciNet  MATH  Google Scholar 

  • Kissita G (2003) Les analyses canoniques généralisées avec tableau de référence généralisé : éléments théoriques et appliqués. PhD thesis, University of Paris Dauphine, France

  • Lohmoller J (1989) Latent variables path modeling with partial least squares. Physica-Verlag, Heidelberg

    Book  MATH  Google Scholar 

  • Martella F, Vicari D, Vichi M (2015) Partitioning predictors in multivariate regression models. Stat Comput 25:261–272

    Article  MathSciNet  MATH  Google Scholar 

  • Preda C, Saporta G (2005) Clusterwise PLS regression on a stochastic process. Comput Stat Data Anal 49:99–108

    Article  MathSciNet  MATH  Google Scholar 

  • Qin S, Valle S, Piovoso M (2001) On unifying multiblock analysis with application to decentralized process monitoring. J Chemom 15:715–742

    Article  Google Scholar 

  • Sarstedt M (2008) A review of recent approaches for capturing heterogeneity in partial least squares path modelling. J Model Manage 3:140–161

    Article  Google Scholar 

  • Schlittgen R, Ringle C, Sarstedt M, Becker JM (2016) Segmentation of PLS path models by iterative reweighted regressions. J Bus Res 69:4583–4592

    Article  Google Scholar 

  • Shao Q, Wu Y (2005) Consistent procedure for determining the number of clusters in regression clustering. J Stat Plan Inference 135:461–476

    Article  MathSciNet  MATH  Google Scholar 

  • Spath H (1979) Clusterwise linear regression. Computing 22:367–373

    Article  MathSciNet  MATH  Google Scholar 

  • Team R (2015) R: a language and environment of statistical computing. http://cran.r-project.org/

  • Tenenhaus A, Tenenhaus M (2011) Regularized generalized canonical correlation analysis. Psychometrika 76:257–284

    Article  MathSciNet  MATH  Google Scholar 

  • Tenenhaus M (1998) La régression PLS. Technip, Paris

    MATH  Google Scholar 

  • Trinchera L (2007) Unobserved heterogeneity in structural equation models: a new approach to latent class detection in PLS path modeling. PhD thesis, University of Naples Federico II

  • Vicari D, Vichi M (2013) Multivariate linear regression for heterogeneous data. J Appl Stat 40:1209–1230

    Article  MathSciNet  Google Scholar 

  • Vinzi V, Lauro C, Amato S (2005) PLS typological regression. In: Vichi M, Monari P, Mignani S, Montanari A (eds) New developments in classification and data analysis. Springer, Berlin, pp 133–140

    Chapter  Google Scholar 

  • Vinzi V, Ringle C, Squillacciotti S, Trinchera L (2007) Capturing and treating unobserved heterogeneity by response based segmentation in PLS path modeling. a comparison of alternative methods by computational experiments. Technical reports, ESSEC Business School, https://www.academia.edu/168969/Capturing_and_Treating_Unobserved_Heterogeneity_by_Response_Based_Segmentation_in_PLS_Path_Modeling._A_Comparison_of_Alternative_Methods_by_Computational_Experiments

  • Vinzi V, Trinchera L, Squillacciotti S, Tenenhaus M (2009) REBUS-PLS: a response-based procedure for detecting unit segments in pls path modeling. Appl Stochastic Models Bus Ind 24:439–458

    Article  MATH  Google Scholar 

  • Vivien M (2002) Approches PLS linéaires et non-linéaires pour la modélisation de multi-tableaux : théorie et applications. PhD thesis, University of Montpellier 1, France

  • Westerhuis J, Coenegracht P (1997) Multivariate modelling of the pharmaceutical two-step process of wet granulation and tableting with multiblock partial least squares. J Chemom 11:379–392

    Article  Google Scholar 

  • Westerhuis J, Smilde A (2001) Deflation in multiblock PLS. J Chemom 15:485–493

    Article  Google Scholar 

  • Westerhuis J, Kourti T, MacGregor J (1998) Analysis of multiblock and hierarchical PCA and PLS model. J Chemom 12:301–321

    Article  Google Scholar 

  • Wold H (1985) Encyclopedia of statistical sciences. In: Kotz S, Johnson N (eds) Partial least squares. Wiley, New York, pp 581–591

    Google Scholar 

  • Wold S (1984) Three PLS algorithms according to SW. Technical reports, Umea University, Sweden

  • Wold S, Martens H, Wold H (1983) The multivariate calibration problem in chemistry solved by the PLS method. Matrix Pencils pp 286–293

Download references

Acknowledgements

The authors are grateful to two anonymous reviewers for their valuable suggestions that greatly improved the clarity and the relevance of this article.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Stéphanie Bougeard.

Appendices

Appendix 1: multiblock PLS

In standard multiblock PLS for the case of a single dataset \(\mathbf {Y}\) to explain, the relationship between \(\mathbf {Y}\) and the K matrices \(\mathbf {X}^{k}\) (stored in \(\mathbf {X}\)) is first modeled by computing a pair of linear combinations \(\mathbf {u}\) and \(\mathbf {t}\)—called components—of the columns of, respectively \(\mathbf {Y}\) and \(\mathbf {X}\) such that these components have maximal covariance (see, e.g., Qin et al. 2001; Abdi and Williams 2012). After this first step—equivalent to a standard PLS model (Qin et al. 2001)—specific components are computed to relate each \(\mathbf {X}^{k}\) to \(\mathbf {Y}\). Formally, MBPLS first implements the following optimization problem

$$\begin{aligned} \displaystyle \delta&= \displaystyle \mathop {{{\mathrm{arg\,max}}}}\limits _{\mathbf {v},\mathbf {w}} \left( {{\mathrm{cov}}}(\mathbf {u}, \mathbf {t}) \right) \quad \text {with} \quad \mathbf {u}=\mathbf {Y}\mathbf {v}, \quad \text {and} \quad \mathbf {t}=\mathbf {X}\mathbf {w} \nonumber \\&\text { under the constraints that} \nonumber \\&\Vert \mathbf {w}\Vert ^2 = \Vert \mathbf {v}\Vert ^2 = 1. \end{aligned}$$
(10)

The solution of this problem is obtained by taking \(\mathbf {v}\) and \(\mathbf {w}\) (called, respectively, the \(\mathbf {Y}\)- and \(\mathbf {X}\)-loadings) as (respectively) the first left and right singular vectors of matrix \(\mathbf {Y}^\textsf {T} \mathbf {X}\) (and the first singular value \(\delta \) gives the thought after maximum of Expression (10)). In a second step, the dependent dataset \(\mathbf {Y}\) is predicted (with a standard linear regression) from the component \(\mathbf {t}\) as

$$\begin{aligned} \widehat{\mathbf {Y}}=\frac{\mathbf {t}\mathbf {t}^\textsf {T}}{\Vert \mathbf {t}\Vert ^2}\mathbf {Y} = \mathbf {t}\mathbf {c}^\textsf {T} \quad \text { with } \quad \mathbf {c} = \frac{\mathbf {Y} ^\textsf {T} \mathbf {t}}{\Vert \mathbf {t}\Vert ^2} \ . \end{aligned}$$
(11)

The matrix \(\widehat{\mathbf {Y}}\) therefore corresponds to the orthogonal projection of \(\mathbf {Y}\) onto the component \(\mathbf {t}\). In MBPLS, after the components and loadings have been found, block loadings (also called partial loadings) are computed (see, e.g., Qin et al. 2001, and Equations (3) and (4) for details) as

$$\begin{aligned} \mathbf {w}^{k} = \frac{(\mathbf {X}^{k}{})^\textsf {T} \mathbf {u}}{\left\| \mathbf {u}^\textsf {T} \mathbf {X}^{k}\right\| } \ , \quad \mathbf {t}^{k} = \mathbf {X}^{k}\mathbf {w}^{k} \quad \text {and} \quad a^{k} = \frac{\mathbf {u}^\textsf {T} \mathbf {t}^{k}}{\sqrt{\displaystyle \sum _{k=1}^K \left( \mathbf {u}^\textsf {T} \mathbf {t}^{k}\right) ^2}}. \end{aligned}$$
(12)

This way of computing the block loading vectors ensures that the global component vector can be obtained as a weighted average of the block vectors, namely

$$\begin{aligned} \mathbf {t} = \sum _{k=1}^K a^{k}\mathbf {t}^{k} \quad \text { with } \quad \sum _{k=1}^K (a^{k}{})^2 = 1 \quad \text { and } \quad \Vert \mathbf {w}^{k}\Vert ^2=1 \ . \end{aligned}$$
(13)

The partial loadings \(\mathbf {w}^{k}\) can be seen as normalized sub-vectors of \(\mathbf {w}\), and this implies that MBPLS can naturally cope with multicollinearity in \(\mathbf {X}_k\) or \(\mathbf {Y}\) and will, therefore, provide stable solutions.

As our regression problem is to get a good prediction of \(\mathbf {Y}\), this dataset is explained with all the variables in \((\mathbf {X}^{1}, \dots , \mathbf {X}^{K})\) (Westerhuis and Smilde 2001). As a consequence, the component-based regression is derived from the global component \(\mathbf {t}\) rather than on block components \(\mathbf {t}^{k}\). Thereafter, plugging Eq. (13) into Eq. (11) shows that matrix \(\mathbf {Y}\) can also be predicted from the partial components as

$$\begin{aligned} \widehat{\mathbf {Y}} = \mathbf {t}\mathbf {c}^\textsf {T} = \sum _{k=1}^K a^{k}\mathbf {t}^{k} \mathbf {c}^\textsf {T}\ . \end{aligned}$$
(14)

Note that when—as it is the case here—the matrix \(\mathbf {Y}\) does not include blocks, the \(\mathbf {X}^{k}\) block loadings are computed after the global loadings have been estimated, and so the block loadings do not depend upon the partition of the explanatory variables in blocks; therefore MBPLS, for the case of a single dependent block \(\mathbf {Y}\), is not a true multiblock method (Westerhuis et al. 1998; Qin et al. 2001; Vivien 2002).

Because one component rarely completely explain the dependent variables, higher order components are often needed. These higher order components are obtained by first removing from the raw data the previous-order solution (a procedure called “deflation”) and then re-iterating the optimization procedure on the deflated data. Because this procedure ensures orthogonality of the components further used in the component-based regression, we choose to deflate the raw data from the global component \(\mathbf {t}\) rather than from the block components \(\mathbf {t}^{k}\). Also, as deflating \(\mathbf {X}\) or \(\mathbf {Y}\) leads to the same prediction (Westerhuis and Smilde 2001), we choose to regress out the effect of the first-order global component from \(\mathbf {X}\). Formally, in our deflation step, \(\mathbf {X}\) is replaced by \(\mathbf {X}^{(2)}\) computed as

$$\begin{aligned} \mathbf {X}^{(2)} = \left( \mathbf {I} - \frac{\mathbf {t}\mathbf {t}^\textsf {T}}{\Vert \mathbf {t}\Vert ^2}\right) \mathbf {X} \ . \end{aligned}$$
(15)

To improve the prediction, \(\mathbf {X}\) is replaced in Eq. (10) by its residual defined in Eq. (15). The process can then be re-iterated to obtain subsequent components. We denote by O the optimal number of components to keep in the model (with \(O \le J\))—O is in general estimated by a cross-validation approach. This deflation step ensures that components (i.e., the vectors \(\mathbf {t}\)) obtained at different steps are orthogonal to each other. Therefore, the predicted dependent dataset can be written according to the global components or according to the block ones

$$\begin{aligned} \widehat{\mathbf {Y}}^{(O)} = \sum _{h=1}^{O} \mathbf {t}^{(h)}(\mathbf {c}^{(h)})^\textsf {T} = \sum _{h=1}^{O} \sum _{k=1}^K a^{k(h)} \mathbf {t}^{k(h)} \left( \mathbf {c}^{(h)}\right) ^\textsf {T} \end{aligned}$$
(16)

with

$$\begin{aligned} \mathbf {c}^{(h)}= \frac{ \mathbf {Y}^\textsf {T} \mathbf {t}^{(h)}}{ \Vert \mathbf {t}^{(h)}\Vert ^2 } \end{aligned}$$
(17)

being the vector of the regression coefficients of \(\mathbf {Y}\) on \(\mathbf {t}^{(h)}\). This last regression step corresponds to the following optimization problem

$$\begin{aligned} \underset{\mathbf {c}}{{{\mathrm{arg\,min}}}} \quad \left\| \mathbf {Y} - \sum _{h=1}^{O} \mathbf {t}^{(h)}(\mathbf {c}^{(h)})^\textsf {T} \right\| ^2 \quad \text {with} \quad \mathbf {t}^{(h)}=\mathbf {X}^{(h-1)} \mathbf {w}^{(h)} \end{aligned}$$
(18)

where \(\mathbf {X}^{(h-1)}\) is the residual of the prediction of \(\mathbf {X}\) from the \(h-1\) previous components \((\mathbf {t}^{(1)}, \dots , \mathbf {t}^{(h-1)})\). Because these components are orthogonal, Expression

$$\begin{aligned} \mathbf {t}^{(h)}=\mathbf {X}^{(h-1)}\mathbf {w}^{(h)} \end{aligned}$$

is equivalent to

$$\begin{aligned} \mathbf {t}^{(h)}=\mathbf {X}{\mathbf {w}^{(h)}}^* \end{aligned}$$

with \({\mathbf {w}^{(h)}}^*\) defined as

$$\begin{aligned} {\mathbf {w}^{(h)}}^*=\prod _{l=1}^{h-1} \left[ \mathbf {I} - \frac{\mathbf {w}^{(l)}(\mathbf {t}^{(l)})^\textsf {T}}{\Vert \mathbf {t}^{(l)}\Vert ^2} \right] \mathbf {w}^{(h)} \end{aligned}$$
(19)

(for proofs see, e.g., Tenenhaus 1998; Wold et al. 1983).

If we define

$$\begin{aligned} {\mathbf {W}^{(O)}}^* = \left[ {\mathbf {w}^{(1)}}^*, \dots , {\mathbf {w}^{(h)}}^*,\dots , {\mathbf {w}^{(O)}}^* \right] \text { and } \mathbf {C}^{(O)} = \left[ {\mathbf {c}^{(1)}}, \dots , {\mathbf {c}^{(h)}},\dots , {\mathbf {c}^{(O)}} \right] \ ,\nonumber \\ \end{aligned}$$
(20)

the optimal prediction of \(\mathbf {Y}\), denoted \(\widehat{\mathbf {Y}}^{(O)}\), can be obtained, in a way analogous to standard multiple linear regression, as

$$\begin{aligned} \widehat{\mathbf {Y}}^{(O)} = \mathbf {X}\mathbf {B}^{(O)} \quad \text {with} \quad \mathbf {B}^{(O)}={\mathbf {W}^{(O)}}^*(\mathbf {C}^{(O)})^\textsf {T}\ . \end{aligned}$$
(21)

Interestingly, rewriting Eq. (21) shows that it can also be obtained as the solution of the following minimization problem

$$\begin{aligned} \underset{\mathbf {c}}{{{\mathrm{arg\,min}}}} \quad \displaystyle \left\| \mathbf {Y}-\sum _{h=1}^{O} \mathbf {t}^{(h)}(\mathbf {c}^{(h)})^\textsf {T} \right\| ^2 \quad \iff \quad \underset{\widehat{\mathbf {Y}} ^{(O)} }{{{\mathrm{arg\,min}}}} \quad \left\| \mathbf {Y}-\widehat{\mathbf {Y}}^{(O)} \right\| ^2 . \end{aligned}$$
(22)

This expression corresponds to a standard least square estimation problem and this indicates, therefore, that the quality of the PLS model can be evaluated like a standard linear regression model using the well-known Root Mean Square Error

$$\begin{aligned} \text {RMSE} = \frac{1}{\sqrt{Q}}\left\| \mathbf {Y}-\widehat{\mathbf {Y}}^{(O)}\right\| . \end{aligned}$$
(23)

Appendix 2: multiblock redundancy analysis

MBRA can be expressed as the solution of the following optimization problem (24)

$$\begin{aligned} \displaystyle \delta = \displaystyle \mathop {{{\mathrm{arg\,max}}}}\limits _{\mathbf {v}, \mathbf {t}^{k}, a^{k}} \left( {{\mathrm{cov}}}^2(\mathbf {u}, \mathbf {t}) \right) \quad \text {with} \quad \mathbf {u}=\mathbf {Y}\mathbf {v}, \quad \mathbf {t} =\sum _{k=1}^K a^{k} \mathbf {t}^{k}, \quad \mathbf {t}^{k}=\mathbf {X}^{k}\mathbf {w}^{k} \end{aligned}$$

under the constraints that

$$\begin{aligned} \displaystyle \sum _{k=1}^K (a^{k})^2 =1 \quad \text {and} \quad \Vert \mathbf {t}^{k}\Vert ^2=\Vert \mathbf {v}\Vert ^2=1 \ . \end{aligned}$$
(24)

It can be shown that the solution of this problem is obtained by taking \(\mathbf {v}\) as the first eigenvector of the matrix

$$\begin{aligned} \sum _{k=1}^K \mathbf {Y}^\textsf {T} \mathbf {P}^{k} \mathbf {Y} \end{aligned}$$
(25)

(see, e.g., Bougeard et al. 2007, 2011a for proofs and details).

In MBRA, block components come from the normalized projections of \(\mathbf {u}\) onto each subspace spanned by the variables of \(\mathbf {X}^{k}\) and are computed as

$$\begin{aligned} \mathbf {t}^{k} = \frac{\mathbf {P}^{k}\mathbf {u}}{\Vert \mathbf {P}^{k}\mathbf {u}\Vert } \quad \text {with} \quad a^{k} = \frac{ \mathbf {u}^\textsf {T} \mathbf {t}^{k}}{ \sqrt{\displaystyle \sum _{k=1}^K \left( \mathbf {u}^\textsf {T} \mathbf {t}^{k} \right) ^2} }. \end{aligned}$$
(26)

In MBRA, the global component is obtained as the weighted sum of the block components, namely

$$\begin{aligned} \mathbf {t} = \sum _{k=1}^K a^{k}\mathbf {t}^{k} \quad \text { with } \quad \sum _{k=1}^K (a^{k}{})^2 = 1 \quad \text { and } \quad \Vert \mathbf {t}^{k}\Vert ^2=1 \ . \end{aligned}$$
(27)

It can be noticed that global as well as block components of MBRA take into account the partition of the explanatory variables in blocks. Furthermore—compared to MBPLSMBRA is more oriented towards the explanation of \(\mathbf {Y}\) but will be less stable in case of multicollinearity within explanatory blocks because it requires matrix inversions \(\left( \text {i.e., } \left( (\mathbf {X}^{k})^\textsf {T} \mathbf {X}^{k}\right) ^{-1} \right) \) as indicated in Eqs. (25) and (26) see, for details, Bougeard et al. (2011a).

As for MBPLS, the effect of the component \(\mathbf {t}\) is regressed out of \(\mathbf {X}\) through the deflation of \(\mathbf {X}\) upon this global component following Eq. (15). Subsequent components are then obtained by replacing matrix \(\mathbf {X}\) in Eq. (24) by its successive residual matrices.

In a second step, the dependent dataset \(\mathbf {Y}\) is predicted using the successive components \((\mathbf {t}^{(1)}, \dots , \mathbf {t}^{(H)})\) and Eqs. (16) and (17) for O—the optimal number of components in the model (in general obtained through a cross-validation procedure).

As for MBPLS, the regression step of MBRA can be interpreted as the solution to the following optimization problem

$$\begin{aligned} \underset{\mathbf {c}}{{{\mathrm{arg\,min}}}} \quad \left\| \mathbf {Y} - \sum _{h=1}^{O} a^{k(h)} \mathbf {t}^{k(h)}(\mathbf {c}^{(h)})^\textsf {T} \right\| ^2 \quad \text {with} \quad \mathbf {t}^{k(h)}=\mathbf {X}^{k(h-1)} \mathbf {w}^{k(h)} \end{aligned}$$
(28)

where \(\mathbf {X}^{k(h-1)}\) is the residual of the prediction of \(\mathbf {X}^{k}\) from the \(h-1\) previous components \((\mathbf {t}^{(1)}, \dots , \mathbf {t}^{(h-1)})\).

Appendix 3: computation times for some representative case studies

See Table 8.

Table 8 Computation times of the clusterwise multiblock algorithm depending on the number of observations N, of explanatory variables J, of components included in the model H, and of clusters G

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bougeard, S., Abdi, H., Saporta, G. et al. Clusterwise analysis for multiblock component methods. Adv Data Anal Classif 12, 285–313 (2018). https://doi.org/10.1007/s11634-017-0296-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11634-017-0296-8

Keywords

Mathematics Subject Classification

Navigation