CSCA: Common and Specific Correspondence Analysis (CSCA) of a set...
In HerveAbdi/PTCA4CATA: Partial Triadic Analysis (PTCA) for Check All That Apply (CATA) Data
CSCA R Documentation
Common and Specific Correspondence Analysis
(CSCA)
of a set of K matched matrices, each of order I*J.
Description
CSCA: implements
Common and Specific Correspondence Analysis
of a set of K
matched matrices, each of order I*J.
Usage
CSCA(brickOfMat, nfact = 3, b = NULL)
Arguments
brickOfMat
an I items by J descriptors by
K blocks (e.g., matrices) suitable
for correspondence analysis (i.e., all non-negative elements).
nfact
(Default = 3) number of factors to keep.
b
(Default = NULL)
a K elements weight vector for the matrices
(should all be positive and sum to 1). When NULL
CSCA computes the weights as the sum of each matrix
divided by the grand total. Note that it is general better
to have pre-normalized the matrices for CSCA
(e.g., with normBrick4PTCA),
so that all matrices have the same weight.
Details
The analysis of the three matrices whose results are
given in
allMatrices.resCA, sumOfMatrices.resCA, and
diffOfMatrices implement an ANOVA like decomposition
of the Inertia such that All = Sum + Difference or
X_K = G +
(X_K - G), with
X_K being the set of matrices,
G being the sum matrix
(which in CA is the barycenter of all the matrices)
and (X_K - G)
being the set of the differences of all the matrices
to their barycenter.
The matrix X_K
is obtained by stacking all the original matrices
on top of each other
(so X_K is an I*K by J matrix);
the
matrix (X_K - G)
is obtained by subtracting from each
matrix in X_K the matrix G
(so X_K - G
is an I*K by J matrix);
the matrix G
is obtained as the sum of all the matrices
in X_K (so G
is a I by J matrix).
The analysis of the matrix G is made with
a standard CA program, but the correspondence analysis
of matrices (X_K - G)
needs a special
CA program because this CA uses the row and column
metrics from G,
X_K uses the same centers as G,
whereas the (X_K - G)
matrix is uncentered
(these analyses are performed
by the function genCA
that allow specific metrics and centers).
For the analysis of X_K and
(X_K - G)
the row factor scores (fi) are
computed from the plain genCA
analysis and the column factor
scores (fj)
are obtained from partial projection using the
correspondence analysis transition formula adapted to blocks
of matrices.
Note that in a two table version
the partial column factor scores will be identical
(and identical to the overall column factor score) and
so in this case, the overal column factor scores can
be plotted.
Note, also, that the two table version
of CSCA could be obtained
from the analysis of the
[X_1 X_2 || X_2 X_1]
circulant matrix
(see Greenacre, 2003).
Value
A list with
1) allMatrices.resCA: results for the analysis of
the whole set of matrices stacked on top of each other;
2) sumOfMatrices.resCA:
results (from ExPosition::epCA) for the analysis of
the sum (i.e., average with CA) of all matrices;
3) diffOfMatrices:
results for the analysis of
the difference of the matrices to their average (from 2);
4) partialProjOnSum:
The projection as supplementary elements of the matrices
onto their average;
and 5) RvCoefficients:
the matrix of Rv-coefficient between the matrices.
# projZonDif.fi
allMatrices is a list
containing a)
fi: the I*K by nfact matrix of the
row factor scores;
b) fj:
the J*nfact*K array
by nfact array of the
column factor scores;
c) Dv: the singular values;
d) eigs: the eigenvalues;
e) tau: the percentage of Inertia; and
f) Inertia the total inertia;
sumOfMatrices is a list storing the output of
the plain correspondence analysis of the
I*J matrix of the sum of matrices as
analyzed by ExPosition::epCA (see help there for
more details).
diffOfMatrices
is a list
containing a)
fi: the I*K by nfact matrix of the
row factor scores; b) fj:
the J*nfact*K array
by nfact array of the
column factor scores;
b) part.fj: A J*L*K
array of partial column factor scores;
c) projZonDif.fi:
An (I*K) by L matrix of the
projection of the original data
onto the specific space (useful to explore the difference
induced by the original data matrices);
d) Dv: the singular values;
e) eigs: the eigenvalues;
f) tau: the percentage of Inertia;
and
g) Inertia the total inertia.
partialProjOnSum is a
list
containing a)
fi: the I*K by nfact matrix of the
(supplementary) row factor scores;
b) fj:
the I*nfact*K array
by nfact array of the (supplementary)
column factor scores.
References
The ideas used here are derived from:
1) Escofier, B. (1983). Analyse de la différence
entre deux mesures définies sur le produit de deux mêmes
ensembles. Les Cahiers de l'Analyse des Données, 8, 325-329.
2) Escofier, B., & Drouet, D. (1983). Analyse des différences
entre plusieurs tableaux de fréquences.
Les Cahiers de l'Analyse des Données, 8, 491-499;
3) Benali, H., & Escofier, B. (1990).
Analyse factorielle lissée et analyse factorielle
des différences locales
Revue de statistique appliquée, 38, 55-76.
4) Greenacre, M. (2003). Singular value decomposition of
matched matrices. Journal of Applied Statistics, 30,
1101-1113; and
5)
Takane Y. (2014). Constrained Principal Component Analysis
and Related Techniques, Boca Raton: CRC Press.
See Also
normBrick4PTCA genPCA
epCA
Examples
## Not run:
if(interactive()){
#EXAMPLE1
}
## End(Not run)
HerveAbdi/PTCA4CATA documentation built on July 17, 2022, 5:41 a.m.
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| CSCA | R Documentation |
Common and Specific Correspondence Analysis (CSCA) of a set of K matched matrices, each of order I*J.
Description
CSCA: implements
Common and Specific Correspondence Analysis
of a set of K
matched matrices, each of order I*J.
Usage
CSCA(brickOfMat, nfact = 3, b = NULL)
Arguments
brickOfMat |
an I items by J descriptors by K blocks (e.g., matrices) suitable for correspondence analysis (i.e., all non-negative elements). |
nfact |
(Default = 3) number of factors to keep. |
b |
(Default = |
Details
The analysis of the three matrices whose results are
given in
allMatrices.resCA, sumOfMatrices.resCA, and
diffOfMatrices implement an ANOVA like decomposition
of the Inertia such that All = Sum + Difference or
X_K = G +
(X_K - G), with
X_K being the set of matrices,
G being the sum matrix
(which in CA is the barycenter of all the matrices)
and (X_K - G)
being the set of the differences of all the matrices
to their barycenter.
The matrix X_K is obtained by stacking all the original matrices on top of each other (so X_K is an I*K by J matrix); the matrix (X_K - G) is obtained by subtracting from each matrix in X_K the matrix G (so X_K - G is an I*K by J matrix); the matrix G is obtained as the sum of all the matrices in X_K (so G is a I by J matrix).
The analysis of the matrix G is made with
a standard CA program, but the correspondence analysis
of matrices (X_K - G)
needs a special
CA program because this CA uses the row and column
metrics from G,
X_K uses the same centers as G,
whereas the (X_K - G)
matrix is uncentered
(these analyses are performed
by the function genCA
that allow specific metrics and centers).
For the analysis of X_K and
(X_K - G)
the row factor scores (fi) are
computed from the plain genCA
analysis and the column factor
scores (fj)
are obtained from partial projection using the
correspondence analysis transition formula adapted to blocks
of matrices.
Note that in a two table version the partial column factor scores will be identical (and identical to the overall column factor score) and so in this case, the overal column factor scores can be plotted.
Note, also, that the two table version of CSCA could be obtained from the analysis of the [X_1 X_2 || X_2 X_1] circulant matrix (see Greenacre, 2003).
Value
A list with
1) allMatrices.resCA: results for the analysis of
the whole set of matrices stacked on top of each other;
2) sumOfMatrices.resCA:
results (from ExPosition::epCA) for the analysis of
the sum (i.e., average with CA) of all matrices;
3) diffOfMatrices:
results for the analysis of
the difference of the matrices to their average (from 2);
4) partialProjOnSum:
The projection as supplementary elements of the matrices
onto their average;
and 5) RvCoefficients:
the matrix of Rv-coefficient between the matrices.
# projZonDif.fi
allMatrices is a list
containing a)
fi: the I*K by nfact matrix of the
row factor scores;
b) fj:
the J*nfact*K array
by nfact array of the
column factor scores;
c) Dv: the singular values;
d) eigs: the eigenvalues;
e) tau: the percentage of Inertia; and
f) Inertia the total inertia;
sumOfMatrices is a list storing the output of
the plain correspondence analysis of the
I*J matrix of the sum of matrices as
analyzed by ExPosition::epCA (see help there for
more details).
diffOfMatrices
is a list
containing a)
fi: the I*K by nfact matrix of the
row factor scores; b) fj:
the J*nfact*K array
by nfact array of the
column factor scores;
b) part.fj: A J*L*K
array of partial column factor scores;
c) projZonDif.fi:
An (I*K) by L matrix of the
projection of the original data
onto the specific space (useful to explore the difference
induced by the original data matrices);
d) Dv: the singular values;
e) eigs: the eigenvalues;
f) tau: the percentage of Inertia;
and
g) Inertia the total inertia.
partialProjOnSum is a
list
containing a)
fi: the I*K by nfact matrix of the
(supplementary) row factor scores;
b) fj:
the I*nfact*K array
by nfact array of the (supplementary)
column factor scores.
References
The ideas used here are derived from:
1) Escofier, B. (1983). Analyse de la différence entre deux mesures définies sur le produit de deux mêmes ensembles. Les Cahiers de l'Analyse des Données, 8, 325-329.
2) Escofier, B., & Drouet, D. (1983). Analyse des différences entre plusieurs tableaux de fréquences. Les Cahiers de l'Analyse des Données, 8, 491-499;
3) Benali, H., & Escofier, B. (1990). Analyse factorielle lissée et analyse factorielle des différences locales Revue de statistique appliquée, 38, 55-76.
4) Greenacre, M. (2003). Singular value decomposition of matched matrices. Journal of Applied Statistics, 30, 1101-1113; and
5) Takane Y. (2014). Constrained Principal Component Analysis and Related Techniques, Boca Raton: CRC Press.
See Also
normBrick4PTCA genPCA
epCA
Examples
## Not run:
if(interactive()){
#EXAMPLE1
}
## End(Not run)
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