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Partial cumulative correspondence analysis

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Abstract

Partial correspondence analysis (Yanai, in: Diday, Escoufier, Lebart, Pagès, Schektman, Thomassone (eds) Data analysis and informatics IV, North-Holland, Amsterdam, pp 193–207, 1986, in: Hayashi, Jambu, Diday, Osumi (eds) Recent developments in clustering and data analysis, Academic Press, Boston, pp 259–266, 1988) has been introduced in statistical literature to eliminate the effects of an ancillary criterion variable on the relationship between two categorical characters. It is well known that partial and classical correspondence analyses do not perform well if one (or both) of the variables forming the contingency table presents an ordinal structure. Cumulative correspondence analysis is a method that considers the information included in the ordinal variable(s). Nevertheless, in this case, a third categorical variable (ancillary) could also influence the existing relation. In this paper, we extend Yanai’s partial approach to cumulative correspondence analysis and, by using suitable orthogonal projectors, we obtain some properties. Finally, we present two real case studies.

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Appendix

Appendix

(A.1):

Let \({\varvec{\Omega }}\) and \({\varvec{\Phi }}\) be given positive definite symmetric matrices of order \((n\times n)\) and \((p \times p)\), respectively. The GSVD of matrix \(\textbf{A}\) is defined as \(\textbf{A}=\textbf{U}{\varvec{\Lambda }}\textbf{V}^T\) where the columns of \(\textbf{U}\) and \(\textbf{V}\) are orthonormalized with respect to \({\varvec{\Omega }}\) and \({\varvec{\Phi }}\) (that is \(\textbf{U}^T\mathbf {\Omega U} = \textbf{I}\) and \(\textbf{V}^T\mathbf {\Phi V} = \textbf{I}\)), respectively, and \({\varvec{\Lambda }}\) is a diagonal and positive definite matrix containing the generalized singular values, ordered from largest to smallest (Greenacre, 1984; Takane & Shibayama, 1991). It is noted as \(\text{ GSVD }(\textbf{A})_{{\varvec{\Omega }},{\varvec{\Phi }}}\) and it can be obtained by means the ordinary SVD as follows. Let \({\varvec{\Omega }}=\textbf{G}_{\varvec{\Omega }}\textbf{G}_{\varvec{\Omega }}^T\) and \({\varvec{\Phi }}=\textbf{G}_{\varvec{\Phi }}\textbf{G}_{\varvec{\Phi }}^T\) be arbitrary square root decomposition of matrices \({\varvec{\Omega }}\) and \({\varvec{\Phi }}\), respectively, and consider the SVD of matrix \(\textbf{G}_{\varvec{\Omega }}^T\textbf{A}\textbf{G}_{\varvec{\Phi }}\) (that is \(\textbf{G}_{\varvec{\Omega }}^T\textbf{A}\textbf{G}_{\varvec{\Phi }}=\tilde{\textbf{U}}\tilde{{\varvec{\Lambda }}}\tilde{\textbf{V}}^T\)) where \(\tilde{\textbf{U}}^T\tilde{\textbf{U}}=\textbf{I}\), \(\tilde{\textbf{V}}^T\tilde{\textbf{V}}=\textbf{I}\) and \(\tilde{{\varvec{\Lambda }}}\) is a diagonal and positive definite matrix. Generalized singular vectors \(\textbf{U}\) and \(\textbf{V}\) are then given by \(\textbf{U}=(\textbf{G}_{\varvec{\Omega }}^T)^{-1}\tilde{\textbf{U}}\) and \(\textbf{V}=(\textbf{G}_{\varvec{\Phi }}^T)^{-1}\tilde{\textbf{V}}\), respectively, with \({\varvec{\Lambda }}=\tilde{{\varvec{\Lambda }}}\). Note that if matrices \({\varvec{\Omega }}\) and \({\varvec{\Phi }}\) are singular then any g-inverse of them could be used (denoted \({\varvec{\Omega }}^-\)) and the solution will be not unique, while if we want to hold the uniqueness then we may use the Moore-Penrose inverse as g-inverse of \({\varvec{\Omega }}\) (denoted \({\varvec{\Omega }}^+\)) (Takane & Hwang, 2002).

(A.2):

Note that \(\textbf{d}=\textbf{Mc}\) and \(\tilde{\textbf{D}}=\textbf{M}-\textbf{d}\textbf{1}_j^T\). Starting from \(\textbf{C}_1\) such that \(T=trace(\textbf{C}_1^T\textbf{C}_1)\), we have the following identities

$$\begin{aligned} \textbf{C}_1= & {} \textbf{W}^{\frac{1}{2}}\tilde{\textbf{D}}\textbf{N}^T\textbf{D}_I^{-\frac{1}{2}}\\= & {} \textbf{W}^{\frac{1}{2}}(\textbf{M}-\textbf{d}\textbf{1}_j^T)(n\times \textbf{P})^T(n\times \textbf{P}_I)^{-\frac{1}{2}}\\= & {} \sqrt{n}\times \textbf{W}^{\frac{1}{2}}(\textbf{M}-\textbf{d}\textbf{1}_j^T) \textbf{P}^T \textbf{P}_I^{-\frac{1}{2}}\\ \tilde{\textbf{C}}_2= & {} \sqrt{n}\times \textbf{C}_2\\= & {} \sqrt{n}\times \textbf{W}^{\frac{1}{2}}(\textbf{M}\textbf{P}^T-\textbf{Mc}\textbf{1}_j^T\textbf{P}^T)\textbf{P}_I^{-\frac{1}{2}}\\= & {} \sqrt{n}\times \textbf{W}^{\frac{1}{2}}\textbf{M}(\textbf{P}^T-\textbf{c}\textbf{r}^T)\textbf{P}_I^{-\frac{1}{2}}\\= & {} \sqrt{n}\times \textbf{W}^{\frac{1}{2}}\textbf{M}(\textbf{P}-\textbf{r}\textbf{c}^T)^T\textbf{P}_I^{-\frac{1}{2}}\\ \tilde{\textbf{C}}_3= & {} \sqrt{n}\times \textbf{C}_3. \end{aligned}$$

such that \(T=trace(\tilde{\textbf{C}}_2^T\tilde{\textbf{C}}_2)=n\times trace(\textbf{C}_2^T\textbf{C}_2)\), and \(T=trace(\tilde{\textbf{C}}_3^T\tilde{\textbf{C}}_3)=n\times trace(\textbf{C}_3^T\textbf{C}_3)\), where \(\textbf{C}_2={} \textbf{W}^{\frac{1}{2}}(\textbf{M}-\textbf{d}\textbf{1}_j^T) \textbf{P}^T \textbf{P}_I^{-\frac{1}{2}}\) and \(\textbf{C}_3={} \textbf{W}^{\frac{1}{2}}\textbf{M}(\textbf{P}-\textbf{r}\textbf{c}^T)^T\textbf{P}_I^{-\frac{1}{2}}\). Finally, consider the transpose of \(\tilde{\textbf{C}}_3\)

$$\begin{aligned} \tilde{\textbf{C}}_3^T= & {} \sqrt{n}\times \textbf{C}_3^T\\= & {} \sqrt{n}\times \textbf{P}_I^{-\frac{1}{2}}(\textbf{P}-\textbf{r}\textbf{c}^T)\textbf{M}^T\textbf{W}^{\frac{1}{2}}\\= & {} \sqrt{n}\times (\textbf{P}_I^{-\frac{1}{2}}\textbf{P}-\textbf{r}^{\frac{1}{2}}\textbf{c}^T)\textbf{M}^T\textbf{W}^{\frac{1}{2}}\\= & {} \sqrt{n}\times (\textbf{P}_I^{-\frac{1}{2}}\textbf{P}-\textbf{P}^{\frac{1}{2}}\textbf{1}_I\textbf{c}^T)\textbf{M}^T\textbf{W}^{\frac{1}{2}}\\= & {} \sqrt{n}\times (\textbf{P}_I^{\frac{1}{2}}\textbf{P}_I^{-1}\textbf{P}-\textbf{P}^{\frac{1}{2}}\textbf{1}_I\textbf{c}^T)\textbf{M}^T\textbf{W}^{\frac{1}{2}}\\= & {} \sqrt{n}\times \textbf{P}_I^{\frac{1}{2}}(\textbf{P}_I^{-1}\textbf{P}-\textbf{1}_I\textbf{c}^T)\textbf{M}^T\textbf{W}^{\frac{1}{2}}\\ \tilde{\textbf{C}}_4= & {} \sqrt{n}\times \textbf{C}_4. \end{aligned}$$

where \(T=trace(\tilde{\textbf{C}}_4\tilde{\textbf{C}}_4^T)=n\times trace(\textbf{C}_4\textbf{C}_4)^T\) and \(\textbf{C}_4=\textbf{P}_I^{\frac{1}{2}}(\textbf{P}_I^{-1}\textbf{P}-\textbf{1}_I\textbf{c}^T)\textbf{M}^T\textbf{W}^{\frac{1}{2}}\).

A.3):

Let L be the Lagrangian function defined as

$$\begin{aligned} \begin{array}{ll} L = (\textbf{a}_{\textbf{X}\tilde{\textbf{K}}}^T\textbf{G}_{\textbf{X}\tilde{\textbf{K}}}^T \textbf{G}_{\hat{\textbf{Y}}\tilde{\textbf{K}}} {\textbf{a}_{\hat{\textbf{Y}}\tilde{\textbf{K}}}} )&{} - \frac{\gamma }{2} (\textbf{a}_{\textbf{X}\tilde{\textbf{K}}}^T {\textbf{G}_{\textbf{X}\tilde{\textbf{K}}}^T\textbf{G}_{\textbf{X}\tilde{\textbf{K}}}} {\textbf{a}_{\textbf{X}\tilde{\textbf{K}}}} - 1)\\ &{} - \frac{\mu }{2} (\textbf{a}_{\hat{\textbf{Y}}\tilde{\textbf{K}}} ^T (\hat{\textbf{M}}^T\hat{\textbf{W}}\hat{\textbf{M}})^- {\textbf{a}_{\hat{\textbf{Y}}\tilde{\textbf{K}}}} - 1) \end{array} \end{aligned}$$

The normal equations are

$$\begin{aligned} \frac{\partial L}{\partial {\textbf{a}_{\textbf{X}\tilde{\textbf{K}}}}}= & {} \textbf{G}_{\textbf{X}\tilde{\textbf{K}}}^T \textbf{G}_{\hat{\textbf{Y}}\tilde{\textbf{K}}} {\textbf{a}_{\hat{\textbf{Y}}\tilde{\textbf{K}}}}-\gamma {\textbf{G}_{\textbf{X}\tilde{\textbf{K}}}^T\textbf{G}_{\textbf{X}\tilde{\textbf{K}}}} {\textbf{a}_{\textbf{X}\tilde{\textbf{K}}}}= 0 \end{aligned}$$
(17)
$$\begin{aligned} \frac{\partial L}{\partial {\textbf{a}_{\hat{\textbf{Y}}\tilde{\textbf{K}}}} }= & {} \textbf{G}_{\hat{\textbf{Y}}\tilde{\textbf{K}}}^T \textbf{G}_{\textbf{X}\tilde{\textbf{K}}} {\textbf{a}_{\textbf{X}\tilde{\textbf{K}}}} -\mu (\hat{\textbf{M}}^T\hat{\textbf{W}}\hat{\textbf{M}})^-{\textbf{a}_{\hat{\textbf{Y}}\tilde{\textbf{K}}}} = 0 \end{aligned}$$
(18)
$$\begin{aligned} 2\frac{\partial L}{\partial \gamma }= & {} - (\textbf{a}_{\textbf{X}\tilde{\textbf{K}}}^T {\textbf{G}_{\textbf{X}\tilde{\textbf{K}}}^T\textbf{G}_{\textbf{X}\tilde{\textbf{K}}}} {\textbf{a}_{\textbf{X}\tilde{\textbf{K}}}} - 1) = 0 \end{aligned}$$
(19)
$$\begin{aligned} 2\frac{\partial L}{\partial \mu }= & {} - (\textbf{a}_{\hat{\textbf{Y}}\tilde{\textbf{K}}} ^T (\hat{\textbf{M}}^T\hat{\textbf{W}}\hat{\textbf{M}})^- {\textbf{a}_{\hat{\textbf{Y}}\tilde{\textbf{K}}}} - 1) = 0 \end{aligned}$$
(20)

Identities \(\lambda =(\textbf{a}_{\textbf{X}\tilde{\textbf{K}}}^T\textbf{G}_{\textbf{X}\tilde{\textbf{K}}}^T \textbf{G}_{\hat{\textbf{Y}}\tilde{\textbf{K}}} {\textbf{a}_{\hat{\textbf{Y}}\tilde{\textbf{K}}}} )=cov(\textbf{t},\textbf{u})=\gamma =\mu \) are obtained by left multiplying (17) with \(\textbf{a}_{\hat{\textbf{X}}\tilde{\textbf{K}}}^T \) and using (19), or by left multiplying (18) with \(\textbf{a}_{\hat{\textbf{Y}}\tilde{\textbf{K}}} ^T\) using (20). In addition we obtain the following transition formulas from Eqs. (18) and (17), respectively

$$\begin{aligned} {\textbf{a}_{\hat{\textbf{Y}}\tilde{\textbf{K}}}}= & {} \frac{1}{\lambda } (\hat{\textbf{M}}^T\hat{\textbf{W}}\hat{\textbf{M}})\textbf{G}_{\hat{\textbf{Y}}\tilde{\textbf{K}}}^T \textbf{G}_{\textbf{X}\tilde{\textbf{K}}} {\textbf{a}_{\textbf{X}\tilde{\textbf{K}}}} \end{aligned}$$
(21)
$$\begin{aligned} {\textbf{a}_{\textbf{X}\tilde{\textbf{K}}}}= & {} \frac{1}{\lambda }({\textbf{G}_{\textbf{X}\tilde{\textbf{K}}}^T\textbf{G}_{\textbf{X}\tilde{\textbf{K}}}})^- \textbf{G}_{\textbf{X}\tilde{\textbf{K}}}^T \textbf{G}_{\hat{\textbf{Y}}\tilde{\textbf{K}}} {\textbf{a}_{\hat{\textbf{Y}}\tilde{\textbf{K}}}} \end{aligned}$$
(22)

The general eigenvalue problem (6) is then achieved by using Eq. (21) in (17) and by left multiplying with \(\textbf{G}_{\textbf{X}\tilde{\textbf{K}}}\). Eigen-system (7) is instead obtained by taking into account the relation \(\textbf{P}_{\textbf{X}\cup \tilde{\textbf{K}}}=\textbf{P}_{\tilde{\textbf{K}}}+\textbf{P}_{\textbf{X}/\tilde{\textbf{K}}}\) in (6) and by left multiplying with \(\textbf{Q}_{\tilde{\textbf{K}}}\), where \(\textbf{P}_{\textbf{X}/\tilde{\textbf{K}}}=\textbf{Q}_{\tilde{\textbf{K}}}\textbf{X}(\textbf{X}^T \textbf{Q}_{\tilde{\textbf{K}}} \textbf{X})^-\textbf{X}^T\textbf{Q}_{\tilde{\textbf{K}}}\). In fact, after rewriting \(\textbf{P}_{\textbf{X}\cup \tilde{\textbf{K}}}\) and left multiplying (6) with \(\textbf{Q}_{\tilde{\textbf{K}}}\), we achieve the following identities

$$\begin{aligned} \begin{array}{rl} \textbf{P}_{\textbf{X}/\tilde{\textbf{K}}} \textbf{G}_{\hat{\textbf{Y}}\tilde{\textbf{K}}} \hat{\textbf{M}}^T\hat{\textbf{W}}\hat{\textbf{M}}^T \textbf{G}_{\hat{\textbf{Y}}\tilde{\textbf{K}}}^T\textbf{G}_{\textbf{X}\tilde{\textbf{K}}}{\textbf{a}_{\textbf{X}\tilde{\textbf{K}}}} =&{} \lambda ^2\textbf{Q}_{\tilde{\textbf{K}}}\textbf{G}_{\textbf{X}\tilde{\textbf{K}}}{\textbf{a}_{\textbf{X}\tilde{\textbf{K}}}}\\ \textbf{P}_{\textbf{X}/\tilde{\textbf{K}}} \left[ \hat{\textbf{Y}}|\tilde{\textbf{K}}\right] \hat{\textbf{M}}^T\hat{\textbf{W}}\hat{\textbf{M}}^T \left[ \hat{\textbf{Y}}|\tilde{\textbf{K}}\right] ^T\left[ \textbf{X}|\tilde{\textbf{K}}\right] {\textbf{a}_{\textbf{X}\tilde{\textbf{K}}}} =&{} \lambda ^2\textbf{Q}_{\tilde{\textbf{K}}}\left[ \textbf{X}|\tilde{\textbf{K}}\right] {\textbf{a}_{\textbf{X}\tilde{\textbf{K}}}}\\ \textbf{P}_{\textbf{X}/\tilde{\textbf{K}}} \left[ \hat{\textbf{Y}}|\tilde{\textbf{K}}\right] \hat{\textbf{M}}^T\hat{\textbf{W}}\hat{\textbf{M}}^T {\left[ \begin{array}{l} \hat{\textbf{Y}}^T\\ \hline \tilde{\textbf{K}}^T \end{array}\right] } \left[ \textbf{X}|\tilde{\textbf{K}}\right] {\textbf{a}_{\textbf{X}\tilde{\textbf{K}}}} =&{} \lambda ^2\textbf{Q}_{\tilde{\textbf{K}}}\left[ \textbf{X}|\tilde{\textbf{K}}\right] {\textbf{a}_{\textbf{X}\tilde{\textbf{K}}}}\\ \left[ \textbf{P}_{\textbf{X}/\tilde{\textbf{K}}} \hat{\textbf{Y}}|\textbf{0}\right] \hat{\textbf{M}}^T\hat{\textbf{W}}\hat{\textbf{M}}^T {\left[ \begin{array}{l} \hat{\textbf{Y}}^T\\ \hline \tilde{\textbf{K}}^T \end{array}\right] } \left[ \textbf{X}|\tilde{\textbf{K}}\right] {\textbf{a}_{\textbf{X}\tilde{\textbf{K}}}} =&{} \lambda ^2\textbf{Q}_{\tilde{\textbf{K}}}\left[ \textbf{X}|\tilde{\textbf{K}}\right] {\textbf{a}_{\textbf{X}\tilde{\textbf{K}}}}\\ \left[ \textbf{P}_{\textbf{X}/\tilde{\textbf{K}}} \hat{\textbf{Y}}\textbf{M}^T\textbf{W}\textbf{M}^T|\textbf{0}\right] {\left[ \begin{array}{l} \hat{\textbf{Y}}^T\\ \hline \tilde{\textbf{K}}^T \end{array}\right] } \left[ \textbf{X}|\tilde{\textbf{K}}\right] {\textbf{a}_{\textbf{X}\tilde{\textbf{K}}}} =&{} \lambda ^2\textbf{Q}_{\tilde{\textbf{K}}}\left[ \textbf{X}|\tilde{\textbf{K}}\right] {\textbf{a}_{\textbf{X}\tilde{\textbf{K}}}}\\ \left[ \textbf{P}_{\textbf{X}/\tilde{\textbf{K}}} \textbf{Y}\textbf{M}^T\textbf{W}\textbf{M}^T|\textbf{0}\right] {\left[ \begin{array}{cc} \textbf{Y}^T\textbf{Q}_{\tilde{\textbf{K}}}\textbf{X}&{}\textbf{Y}^T\textbf{Q}_{\tilde{\textbf{K}}}\tilde{\textbf{K}}\\ \tilde{\textbf{K}}^T\textbf{X}&{}\tilde{\textbf{K}}^T\tilde{\textbf{K}} \end{array}\right] } {\textbf{a}_{\textbf{X}\tilde{\textbf{K}}}} =&{} \lambda ^2\left[ \textbf{Q}_{\tilde{\textbf{K}}} \textbf{X}|\textbf{0}\right] {\textbf{a}_{\textbf{X}\tilde{\textbf{K}}}}\\ \left[ \textbf{P}_{\textbf{X}/\tilde{\textbf{K}}} \textbf{Y}\textbf{M}^T\textbf{W}\textbf{M}^T|\textbf{0}\right] {\left[ \begin{array}{cc} \textbf{Y}^T\textbf{Q}_{\tilde{\textbf{K}}}\textbf{X}&{}\textbf{0}\\ \tilde{\textbf{K}}^T\textbf{X}&{}\tilde{\textbf{K}}^T\tilde{\textbf{K}} \end{array}\right] } {\textbf{a}_{\textbf{X}\tilde{\textbf{K}}}} =&{} \lambda ^2\left[ \textbf{Q}_{\tilde{\textbf{K}}} \textbf{X}|\textbf{0}\right] {\textbf{a}_{\textbf{X}\tilde{\textbf{K}}}}\\ \left[ \textbf{P}_{\textbf{X}/\tilde{\textbf{K}}} \textbf{Y}\textbf{M}^T\textbf{W}\textbf{M}^T \textbf{Y}^T\textbf{Q}_{\tilde{\textbf{K}}}\textbf{X}|\textbf{0} \right] {\textbf{a}_{\textbf{X}\tilde{\textbf{K}}}} =&{} \lambda ^2\left[ \textbf{Q}_{\tilde{\textbf{K}}} \textbf{X}|\textbf{0}\right] {\textbf{a}_{\textbf{X}\tilde{\textbf{K}}}}\\ \textbf{P}_{\textbf{X}/\tilde{\textbf{K}}} \textbf{Y}\textbf{M}^T\textbf{W}\textbf{M}^T \textbf{Y}^T\textbf{Q}_{\tilde{\textbf{K}}}\textbf{X} \textbf{a}_{\textbf{X}} =&{} \lambda ^2\textbf{Q}_{\tilde{\textbf{K}}} \textbf{X}\textbf{a}_{\textbf{X}}\\ \textbf{Q}_{\tilde{\textbf{K}}}\textbf{X}(\textbf{X}^T \textbf{Q}_{\tilde{\textbf{K}}} \textbf{X})^-\textbf{X}^T\textbf{Q}_{\tilde{\textbf{K}}}\textbf{Y}\textbf{M}^T\textbf{W}\textbf{M}^T \textbf{Y}^T\textbf{Q}_{\tilde{\textbf{K}}}\textbf{X} \textbf{a}_{\textbf{X}} =&{} \lambda ^2\textbf{Q}_{\tilde{\textbf{K}}} \textbf{X}\textbf{a}_{\textbf{X}}\\ (\textbf{X}^T \textbf{Q}_{\tilde{\textbf{K}}} \textbf{X})^-\textbf{X}^T\textbf{Q}_{\tilde{\textbf{K}}}\textbf{Y}\textbf{M}^T\textbf{W}\textbf{M}^T \textbf{Y}^T\textbf{Q}_{\tilde{\textbf{K}}}\textbf{X} \textbf{a}_{\textbf{X}} =&{} \lambda ^2\textbf{a}_{\textbf{X}}\\ \end{array} \end{aligned}$$

where last identy is obtained left-multiplying the previuos one with \((\textbf{X}^T \textbf{Q}_{\tilde{\textbf{K}}} \textbf{X})^-\textbf{X}^T\).

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Amenta, P., D’Ambra, A. & Lucadamo, A. Partial cumulative correspondence analysis. Ann Oper Res 342, 1495–1527 (2024). https://doi.org/10.1007/s10479-022-05141-0

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