distatis: 3-Way MDS based on the "STATIS" optimization procedure.
In DistatisR: DiSTATIS Three Way Metric Multidimensional Scaling
distatis R Documentation
3-Way MDS based on the "STATIS" optimization
procedure.
Description
distatis: Implements the DISTATIS
method which is a 3-way generalization of
metric multidimensional scaling
(a.k.a. classical MDS or principal coordinate analysis).
Usage
distatis(
LeCube2Distance,
Norm = "MFA",
Distance = TRUE,
double_centering = TRUE,
RV = TRUE,
nfact2keep = 3,
compact = FALSE
)
Arguments
LeCube2Distance
an "observations
* observations
* distance matrices" array of dimensions
I*I*K.
Each of the K "slices" is a I*I square
distance (or covariance) matrix describing the
I observations.
Norm
Type of normalization
used for each cross-product matrix derived
from the distance (or covariance) matrices.
Current options are NONE
(do nothing), SUMPCA (normalize by the total inertia)
or MFA (default) that normalizes each matrix so
that its first eigenvalue is equal to one
or NUCLEAR (i.e., the of the squarae root of the
eigenvalues).
Distance
if TRUE (default)
the matrices are distance matrices, FALSE
the matrices are treated as positive semi-definite matrices
(e.g., scalar products,
covariance, or correlation matrices).
double_centering
if TRUE
(default) the matrices are double-centered
(should always be used for distances).
if FALSE the matrices
will not be double centered
(note that these matrices
should be semi positive definite matrices such as,
for example,
covariance matrices).
RV
if TRUE (default)
we use the Rv coefficient to
compute the weights,
if FALSE
we use the matrix scalar product.
nfact2keep
(default: 3) Number of factors
to keep for the computation of the
factor scores of the observations.
compact
if FALSE (default),
distatis provides detailed output, if
TRUE, distatis sends back
only the alpha weights
(this option is used to make the
bootstrap routine
BootFromCompromise more
computationally efficient).
Details
distatis takes
as input a set of K distance matrices
(or positive semi-definite matrices such as scalar products,
covariance, or correlation matrices)
describing a set of I observations.
From this set of matrices distatis
computes: (1) a set of
factor scores that describes the similarity structure
of the K distance
matrices (e.g., what distance matrices describe the
observations in the same
way, what distance matrices differ from each other)
(2) a set of factor
scores (called the compromise factor scores)
that best describes
the similarity structure of the I observations
and (3)
I
sets of
partial factor scores that show how
each individual distance matrix "sees"
the compromise space.
distatis computes the compromise as an optimum
linear combination of the cross-product matrices
associated to each distance
(or positive positive semi-definite)
matrix.
distatis can also be applied to a set of
scalar products, covariance, or correlation
matrices.
DISTATIS is part of the
STATIS family.
It is often used to analyze
the results of sorting tasks.
Value
distatis sends back the results
via two lists:
res.Cmat
and res.Splus.
Note that items with a * are the only ones sent back
when using the compact = TRUE option.
res.Cmat
Results for the between
distance matrices analysis.
-
res.Cmat$C
The I*I C matrix
of scalar products (or Rv between distance matrices).
-
res.Cmat$vectors The eigenvectors of the C matrix
-
res.Cmat$alpha * The alpha weights
-
res.Cmat$value The eigenvalues of the C matrix
-
res.Cmat$G The factor scores for the C matrix
-
res.Cmat$ctr The contributions for res.Cmat$G,
-
res.Cmat$cos2 The squared cosines for res.Cmat$G
-
res.Cmat$d2 The squared
Euclidean distance for res.Cmat$G.
res.Splus
Results for the between observation analysis.
-
res.Splus$SCP an I*I*K array.
Contains
the (normalized if needed)
cross product matrices corresponding to the
distance matrices.
-
res.Splus$Splus * The compromise
(optimal linear
combination of the SCP's').
-
res.Splus$eigValues *
The eigenvalues of the compromise).
-
res.Splus$eigVectors *
The eigenvectors of the compromise).
-
res.Splus$tau * The percentage
of explained inertia of the eigenValues).
-
res.Splus$ProjectionMatrix The
projection matrix used to compute factor
scores and partial factor scores.
-
res.Splus$F The factor scores for the observations.
-
res.Splus$ctr The contributions for res.Cmat$F.
-
res.Splus$cos2 The squared cosines for res.Cmat$F.
-
res.Splust$d2 The squared
Euclidean distance for res.Cmat$F.
-
res.Splus$PartialF an
I*nf2keep*K array.
Contains the partial factors for the distance
matrices.
Author(s)
Hervé Abdi
#@seealso GraphDistatisAll
GraphDistatisBoot
#GraphDistatisCompromise
# GraphDistatisPartial
#GraphDistatisRv DistanceFromSort
#BootFactorScores BootFromCompromise
#as help,
References
Abdi, H., Valentin, D., O'Toole, A.J., & Edelman, B.
(2005).
DISTATIS: The analysis of multiple distance matrices.
Proceedings of
the IEEE Computer Society:
International Conference on Computer Vision and
Pattern Recognition. (San Diego, CA, USA). pp. 42–47.
Abdi, H., Valentin, D., Chollet, S., & Chrea, C. (2007).
Analyzing
assessors and products in sorting tasks:
DISTATIS, theory and applications.
Food Quality and Preference, 18, 627–640.
Abdi, H., Dunlop, J.P., & Williams, L.J. (2009).
How to compute reliability
estimates and display confidence and tolerance
intervals for pattern
classifiers using the Bootstrap and 3-way multidimensional scaling
(DISTATIS).
NeuroImage, 45, 89–95.
Abdi, H., Williams, L.J., Valentin, D., &
Bennani-Dosse, M. (2012).
STATIS
and DISTATIS: Optimum multi-table principal component
analysis and three way
metric multidimensional scaling.
Wiley Interdisciplinary Reviews:
Computational Statistics, 4, 124–167.
The R_V coefficient is described in
Abdi, H. (2007). RV coefficient and congruence coefficient.
In N.J. Salkind
(Ed.): Encyclopedia of Measurement and Statistics.
Thousand Oaks
(CA): Sage. pp. 849–853.
Abdi, H. (2010). Congruence: Congruence coefficient,
RV coefficient, and
Mantel Coefficient. In N.J. Salkind, D.M., Dougherty,
& B. Frey (Eds.):
Encyclopedia of Research Design.
Thousand Oaks (CA): Sage. pp.
222–229.
(These papers are available from
https://personal.utdallas.edu/~herve/)
Examples
# 1. Load the DistAlgo data set
# (available from the DistatisR package).
data(DistAlgo)
# DistAlgo is a 6*6*4 Array (face*face*Algorithm)
#------------------------------------------------------------------
# 2. Call the DISTATIS routine with the array
# of distance (DistAlgo) as parameter
DistatisAlgo <- distatis(DistAlgo)
DistatisR documentation built on Dec. 5, 2022, 9:05 a.m.
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| distatis | R Documentation |
3-Way MDS based on the "STATIS" optimization procedure.
Description
distatis: Implements the DISTATIS
method which is a 3-way generalization of
metric multidimensional scaling
(a.k.a. classical MDS or principal coordinate analysis).
Usage
distatis( LeCube2Distance, Norm = "MFA", Distance = TRUE, double_centering = TRUE, RV = TRUE, nfact2keep = 3, compact = FALSE )
Arguments
LeCube2Distance |
an "observations * observations * distance matrices" array of dimensions I*I*K. Each of the K "slices" is a I*I square distance (or covariance) matrix describing the I observations. |
Norm |
Type of normalization
used for each cross-product matrix derived
from the distance (or covariance) matrices.
Current options are |
Distance |
if |
double_centering |
if |
RV |
if |
nfact2keep |
(default: |
compact |
if |
Details
distatis takes
as input a set of K distance matrices
(or positive semi-definite matrices such as scalar products,
covariance, or correlation matrices)
describing a set of I observations.
From this set of matrices distatis
computes: (1) a set of
factor scores that describes the similarity structure
of the K distance
matrices (e.g., what distance matrices describe the
observations in the same
way, what distance matrices differ from each other)
(2) a set of factor
scores (called the compromise factor scores)
that best describes
the similarity structure of the I observations
and (3)
I
sets of
partial factor scores that show how
each individual distance matrix "sees"
the compromise space.
distatis computes the compromise as an optimum
linear combination of the cross-product matrices
associated to each distance
(or positive positive semi-definite)
matrix.
distatis can also be applied to a set of
scalar products, covariance, or correlation
matrices.
DISTATIS is part of the STATIS family. It is often used to analyze the results of sorting tasks.
Value
distatis sends back the results
via two lists:
res.Cmat
and res.Splus.
Note that items with a * are the only ones sent back
when using the compact = TRUE option.
res.Cmat |
Results for the between distance matrices analysis. |
-
res.Cmat$CThe I*I C matrix of scalar products (or Rv between distance matrices). -
res.Cmat$vectorsThe eigenvectors of the C matrix -
res.Cmat$alpha* The alpha weights -
res.Cmat$valueThe eigenvalues of the C matrix -
res.Cmat$GThe factor scores for the C matrix -
res.Cmat$ctrThe contributions forres.Cmat$G, -
res.Cmat$cos2The squared cosines forres.Cmat$G -
res.Cmat$d2The squared Euclidean distance forres.Cmat$G.
res.Splus |
Results for the between observation analysis. |
-
res.Splus$SCPan I*I*K array. Contains the (normalized if needed) cross product matrices corresponding to the distance matrices. -
res.Splus$Splus* The compromise (optimal linear combination of the SCP's'). -
res.Splus$eigValues* The eigenvalues of the compromise). -
res.Splus$eigVectors* The eigenvectors of the compromise). -
res.Splus$tau* The percentage of explained inertia of the eigenValues). -
res.Splus$ProjectionMatrixThe projection matrix used to compute factor scores and partial factor scores. -
res.Splus$FThe factor scores for the observations. -
res.Splus$ctrThe contributions forres.Cmat$F. -
res.Splus$cos2The squared cosines forres.Cmat$F. -
res.Splust$d2The squared Euclidean distance forres.Cmat$F. -
res.Splus$PartialFan I*nf2keep*K array. Contains the partial factors for the distance matrices.
Author(s)
Hervé Abdi
#@seealso GraphDistatisAll
GraphDistatisBoot
#GraphDistatisCompromise
# GraphDistatisPartial
#GraphDistatisRv DistanceFromSort
#BootFactorScores BootFromCompromise
#as help,
References
Abdi, H., Valentin, D., O'Toole, A.J., & Edelman, B. (2005). DISTATIS: The analysis of multiple distance matrices. Proceedings of the IEEE Computer Society: International Conference on Computer Vision and Pattern Recognition. (San Diego, CA, USA). pp. 42–47.
Abdi, H., Valentin, D., Chollet, S., & Chrea, C. (2007). Analyzing assessors and products in sorting tasks: DISTATIS, theory and applications. Food Quality and Preference, 18, 627–640.
Abdi, H., Dunlop, J.P., & Williams, L.J. (2009). How to compute reliability estimates and display confidence and tolerance intervals for pattern classifiers using the Bootstrap and 3-way multidimensional scaling (DISTATIS). NeuroImage, 45, 89–95.
Abdi, H., Williams, L.J., Valentin, D., & Bennani-Dosse, M. (2012). STATIS and DISTATIS: Optimum multi-table principal component analysis and three way metric multidimensional scaling. Wiley Interdisciplinary Reviews: Computational Statistics, 4, 124–167.
The R_V coefficient is described in
Abdi, H. (2007). RV coefficient and congruence coefficient. In N.J. Salkind (Ed.): Encyclopedia of Measurement and Statistics. Thousand Oaks (CA): Sage. pp. 849–853.
Abdi, H. (2010). Congruence: Congruence coefficient, RV coefficient, and Mantel Coefficient. In N.J. Salkind, D.M., Dougherty, & B. Frey (Eds.): Encyclopedia of Research Design. Thousand Oaks (CA): Sage. pp. 222–229.
(These papers are available from https://personal.utdallas.edu/~herve/)
Examples
# 1. Load the DistAlgo data set # (available from the DistatisR package). data(DistAlgo) # DistAlgo is a 6*6*4 Array (face*face*Algorithm) #------------------------------------------------------------------ # 2. Call the DISTATIS routine with the array # of distance (DistAlgo) as parameter DistatisAlgo <- distatis(DistAlgo)
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