Bus data

Description

Three-way data about the process of learning to read of seven first-grade children tested weekly (from week 3 to 47, but weeks 10, 19, 20, 29, 35, 36, 39, 43 were holidays and, thus, data on 37 weeks) with five different tests.

Usage

data(Bus)

Format

A matrix with 7 rows and 185 (5x37) columns.
The rows refer to the pupils.
The columns refer to the combinations of tests and weeks with the tests nested within the weeks.
The matrix contains the frontal slices next to each other of the original array.
The meanings and the ranges of the tests are as follows:
L: letter knowledge test (scores in 0-47);
P: regular orthographic short words (scores in 0-10);
Q: regular orthographic long words (scores in 0-10);
S: regular orthographic long and short words within context (scores in 0-15);
R: irregular orthographic long and short words (scores in 0-15).

Details

In the literature the Bus data have been analyzed by Tucker3 (see Kroonenberg, 1983; Timmerman, 2001). There is consensus on normalizing the data so to eliminate artificial differences among ranges of tests. Different centering options and numbers of extracted components have been chosen. Specifically, Kroonenberg (1983) suggests averaging over pupils and tests for each time occasions and extracting two components for every mode. Timmerman (2001) suggests to apply Tucker3 to the normalized data with two components for pupils and time occasions and one component for tests.

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

References

P.M. Kroonenberg (1983). Three-mode Principal Component Analysis. Theory and Applications. DSWO Press, Leiden.
M.E. Timmerman (2001). Component Analysis of Multisubject Multivariate Longitudinal Data. Ph.D. Thesis, University of Groningen.

Interactive Candecomp/Parafac analysis

Description

Detects the underlying structure of a three-way array according to the Candecomp/Parafac (CP) model.

Usage

 CP(data,laba,labb,labc)

Arguments

Value

A list including the following components:

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

References

J.D. Carroll and J.J. Chang (1970). Analysis of individual differences in multidimensional scaling via an N-way generalization of 'Eckart-Young' decomposition. Psychometrika 35:283–319.
P. Giordani, H.A.L. Kiers, M.A. Del Ferraro (2014). Three-way component analysis using the R package ThreeWay. Journal of Statistical Software 57(7):1–23. http://www.jstatsoft.org/v57/i07/.
R.A. Harshman (1970). Foundations of the Parafac procedure: models and conditions for an 'explanatory' multi-mode factor analysis. UCLA Working Papers in Phonetics 16:1–84.
P.M. Kroonenberg (2008). Applied Multiway Data Analysis. Wiley, New Jersey.

See Also

T3, T2, T1

Examples

data(TV)
TVdata=TV[[1]]
labSCALE=TV[[2]]
labPROGRAM=TV[[3]]
labSTUDENT=TV[[4]]
# permutation of the modes so that the A-mode refers to students
TVdata <- permnew(TVdata, 16, 15, 30)
TVdata <- permnew(TVdata, 15, 30, 16)
## Not run: 
# interactive CP analysis
TVcp <- CP(TVdata, labSTUDENT, labSCALE, labPROGRAM)
# interactive CP analysis (when labels are not available)
TVcp <- CP(TVdata)

## End(Not run)

Plot fit of Candecomp/Parafac

Description

Plots fits against numbers of dimensions, with S as labels and fits against number of effective paramaters.

Usage

CPdimensionalityplot(A, n, m, p)

Arguments

Note

A is usually the first and fourth columns of the output of DimSelector.
The number of effective parameters in a Candecomp/Parafac analysis is discussed in Weesie and Van Houwelingen (1983).

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

References

E. Ceulemans \& H.A.L. Kiers (2006). Selecting among three-mode principal component models of different types and complexities: A numerical convex hull based method. British Journal of Mathematical and Statistical Psychology 59:133–150.
J. Weesie \& H. Van Houwelingen (1983). GEPCAM users' manual (first draft). Utrecht, The Netherlands: Institute of Mathematical Statistics, State University of Utrecht.

See Also

CP, DimSelector

Examples

data(TV)
TVdata=TV[[1]]
# permutation of the modes so that the A-mode refers to students
TVdata <- permnew(TVdata, 16, 15, 30)
TVdata <- permnew(TVdata, 15, 30, 16)
# Fit values of CP with different numbers of components (from 1 to 5)
FitCP <- CPrunsFit(TVdata, 30, 16, 15, 5)
OutCP <- FitCP[,c(1,4)]
CPdimensionalityplot(OutCP, 30, 16, 15)

Fit of each entity per mode

Description

Computation of fit contributions.

Usage

 CPfitpartitioning(Xprep, n, m, p, A, B, C, laba, labb, labc)

Arguments

Value

A list including the following components:

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

See Also

CP

Examples

data(TV)
TVdata=TV[[1]]
labSCALE=TV[[2]]
labPROGRAM=TV[[3]]
labSTUDENT=TV[[4]]
# permutation of the modes so that the A-mode refers to students
TVdata <- permnew(TVdata, 16, 15, 30)
TVdata <- permnew(TVdata, 15, 30, 16)
# CP solution
TVcp <- CPfuncrep(TVdata, 30, 16, 15, 2, 1, 1, 1, 0, 1e-6, 10000)
# Fitpartitioning of the CP solution
FitCP <-  CPfitpartitioning(TVdata, 30, 16, 15, TVcp$A, TVcp$B, TVcp$C, 
 labSTUDENT, labSCALE, labPROGRAM)
# Fitpartitioning of the CP solution (when labels are not available)
FitCP <-  CPfitpartitioning(TVdata, 30, 16, 15, TVcp$A, TVcp$B, TVcp$C)

Algorithm for the Candecomp/Parafac (CP) model

Description

Alternating Least Squares algorithm for the minimization of the Candecomp/Parafac loss function.

Usage

 CPfunc(X, n, m, p, r, ort1, ort2, ort3, start, conv, maxit, A, B, C)

Arguments

Value

A list including the following components:

Note

The loss function to be minimized is sum(k)|| X(k) - A D(k) B' ||^2, where D(k) is a diagonal matrix holding the k-th row of C.
CPfunc is the same as CPfuncrep except that all printings are available.

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

References

R.A. Harshman (1970). Foundations of the Parafac procedure: models and conditions for an ‘explanatory’ multi-mode factor analysis. UCLA Working Papers in Phonetics 16:1–84.

See Also

CP, CPfuncrep

Examples

data(TV)
TVdata=TV[[1]]
# permutation of the modes so that the A-mode refers to students
TVdata <- permnew(TVdata, 16, 15, 30)
TVdata <- permnew(TVdata, 15, 30, 16)
# unconstrained CP solution using two components 
# (rational starting point by SVD [start=0])
TVcp <- CPfunc(TVdata, 30, 16, 15, 2, 1, 1, 1, 0, 1e-6, 10000)
# constrained CP solution using two components with orthogonal A-mode 
# component matrix (rational starting point by SVD [start=0])
TVcp <- CPfunc(TVdata, 30, 16, 15, 2, 2, 1, 1, 0, 1e-6, 10000)
# constrained CP solution using two components with orthogonal A-mode 
# component matrix and zero correlated C-mode component matrix 
# (rational starting point by SVD [start=0])
TVcp <- CPfunc(TVdata, 30, 16, 15, 2, 2, 1, 3, 0, 1e-6, 10000)
# unconstrained CP solution using two components 
# (random orthonormalized starting point [start=1])
TVcp <- CPfunc(TVdata, 30, 16, 15, 2, 1, 1, 1, 1, 1e-6, 10000)
# unconstrained CP solution using two components (user starting point [start=2])
TVcp <- CPfunc(TVdata, 30, 16, 15, 2, 1, 1, 1, 2, 1e-6, 10000, 
 matrix(rnorm(30*2),nrow=30), matrix(rnorm(16*2),nrow=16), 
 matrix(rnorm(15*2),nrow=15))

Algorithm for the Candecomp/Parafac (CP) model

Description

Alternating Least Squares algorithm for the minimization of the Candecomp/Parafac loss function.

Usage

 CPfuncrep(X, n, m, p, r, ort1, ort2, ort3, start, conv, maxit, A, B, C)

Arguments

Value

A list including the following components:

Note

The loss function to be minimized is sum(k)|| X(k) - A D(k) B' ||^2, where D(k) is a diagonal matrix holding the k-th row of C.
CPfuncrep is the same as CPfunc except that all printings are suppressed. Thus, CPfuncrep can be helpful for simulation experiments.

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

References

R.A. Harshman (1970). Foundations of the Parafac procedure: models and conditions for an ‘explanatory’ multi-mode factor analysis. UCLA Working Papers in Phonetics 16:1–84.

See Also

CP, CPfunc

Examples

data(TV)
TVdata=TV[[1]]
# permutation of the modes so that the A-mode refers to students
TVdata <- permnew(TVdata, 16, 15, 30)
TVdata <- permnew(TVdata, 15, 30, 16)
# unconstrained CP solution using two components 
# (rational starting point by SVD [start=0])
TVcp <- CPfuncrep(TVdata, 30, 16, 15, 2, 1, 1, 1, 0, 1e-6, 10000)
# constrained CP solution using two components with orthogonal A-mode  
# component matrix (rational starting point by SVD [start=0])
TVcp <- CPfuncrep(TVdata, 30, 16, 15, 2, 2, 1, 1, 0, 1e-6, 10000)
# constrained CP solution using two components with orthogonal A-mode 
# component matrix and zero correlated C-mode component matrix 
# (rational starting point by SVD [start=0])
TVcp <- CPfuncrep(TVdata, 30, 16, 15, 2, 2, 1, 3, 0, 1e-6, 10000)
# unconstrained CP solution using two components 
# (random orthonormalized starting point [start=1])
TVcp <- CPfuncrep(TVdata, 30, 16, 15, 2, 1, 1, 1, 1, 1e-6, 10000)
# unconstrained CP solution using two components (user starting point [start=2])
TVcp <- CPfuncrep(TVdata, 30, 16, 15, 2, 1, 1, 1, 2, 1e-6, 10000, 
 matrix(rnorm(30*2),nrow=30), matrix(rnorm(16*2),nrow=16), 
 matrix(rnorm(15*2),nrow=15))

Candecomp/Parafac solutions

Description

Computes all the Candecomp/Parafac solutions (CP) with r (from 1 to maxC) components.

Usage

	CPrunsFit(X, n, m, p, maxC)

Arguments

Value

Note

The structure of out is consistent with Tucker models. In CP, the first and forth columns are sufficient for choosing the optimal number of components.

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

References

H.A.L. Kiers (1991). Hierarchical relations among three-way methods. Psychometrika 56:449–470.

See Also

DimSelector, LineCon, CP

Examples

data(TV)
TVdata=TV[[1]]
# permutation of the modes so that the A-mode refers to students
TVdata <- permnew(TVdata, 16, 15, 30)
TVdata <- permnew(TVdata, 15, 30, 16)
# Fit values of CP with different numbers of components (from 1 to 5)
FitCP <- CPrunsFit(TVdata, 30, 16, 15, 5)

Columnwise centering of a matrix

Description

Computation of a columnwise centered version of a matrix.

Usage

 Cc(A)

Arguments

Value

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

See Also

nrm2

Examples

X <- matrix(rnorm(6*3),ncol=3)
Y <- Cc(X)
apply(Y,2,mean)

Convex Hull procedure

Description

Selects among three-mode principal component models of different complexities.

Usage

 DimSelector(out, n, m, p, model)

Arguments

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

References

E. Ceulemans and H.A.L. Kiers (2006). Selecting among three-mode principal component models of different types and complexities: A numerical convex hull based method. British Journal of Mathematical and Statistical Psychology 59:133–150.
J. Weesie and H. Van Houwelingen (1983). GEPCAM users' manual (first draft). Utrecht, The Netherlands: Institute of Mathematical Statistics, State University of Utrecht.

See Also

LineCon, T3runsApproxFit T2runsApproxFit T1runsFit CPrunsFit

Examples

data(Bus)
# Analysis on T3 with different numbers of components (from 1 to 4 for the A-mode,
# from 1 to 3 for the B-mode, from 1 to 5 for the C-mode)
FitT3 <- T3runsApproxFit(Bus,7,5,37,4,4,4)
T3opt <- DimSelector(FitT3,7,5,37,2)

Kinship terms data

Description

Three-way proximity data about 15 kinship terms produced by 6 groups of subjects.

Usage

data(Kinship)

Format

An array of order 15 x 15 x 6.
The A-mode and B-mode entities are the kinship terms (Aunt, Brother, Cousin, Daughter, Father, Granddaughter, Grandfather, Grandmother, Grandson, Mother, Nephew, Niece, Sister, Son, Uncle).
The C-mode entities are groups of subjects (First female, Second female, First male, Second male, Single female, Single male).

Details

The original data have been introduced by Rosenborg \& Kim (1975). The data were collected by asking to 6 groups of subjects to produce a partition of 15 kinship terms. Two groups (Single female and Single male) were composed by 85 male and 85 female college students, respectively, and provided a single partition. Two additional groups of, respectively, 80 male and 80 female students produced two partitions each (First female, Second female, First male, Second male). In fact, they were informed in advance that, after making the first partition, they should give a new partition of the kinship terms using a different basis of meaning. The array contains similarities. For every group of subjects, the numbers of times in which the kinship terms were grouped together are given.

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

References

S. Rosenborg \& M.P. Kim (1975). The method of sorting as a data-gathering procedure in multivariate research. Multivariate Behavioral Research 10:489–502.

Examples

data(Kinship)
## The labels are in the data array
laba <- dimnames(Kinship)[[1]]
labb <- dimnames(Kinship)[[2]]
labc <- dimnames(Kinship)[[3]]
## Candecomp/Parafac analysis
## Not run: 
CP(Kinship,laba,labb,labc)

## End(Not run)

Middle point location

Description

Checks whether the middle point is located below or on the line connecting its neighbors.

Usage

 LineCon(f1, f2, f3, fp1, fp2, fp3)

Arguments

Value

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

References

E. Ceulemans and H.A.L. Kiers (2006). Selecting among three-mode principal component models of different types and complexities: A numerical convex hull based method. British Journal of Mathematical and Statistical Psychology 59:133–150.
J. Weesie and H. Van Houwelingen (1983). GEPCAM users' manual (first draft). Utrecht, The Netherlands: Institute of Mathematical Statistics, State University of Utrecht.

See Also

DimSelector

Examples

data(Bus)
# T2-AB with 1 component for the A- and B-mode
FitBusT2AB11 <- T2funcrep(Bus, 7, 5, 37, 1, 1, 37, 0, 1e-6,1)$fp
# T2-AB with 2 components for the A-mode and 1 component for the B-mode
FitBusT2AB21 <- T2funcrep(Bus, 7, 5, 37, 2, 1, 37, 0, 1e-6, 1)$fp
# T2-AB with 1 component for the A-mode and 2 components for the B-mode
# T2-AB with 1 component for the A-mode and 2 components for the B-mode
# FitBusT2AB21>FitBusT2AB12
# T2-AB with 2 components for the A- and B-mode
FitBusT2AB22 <- T2funcrep(Bus, 7, 5, 37, 2, 2, 37, 0, 1e-6,1)$fp
# number of effective parameters n x r1 + m x r2 + r1 x r2 x p - r1^2 - r2^2
nepT2AB11 <- 47
nepT2AB21 <- 88
nepT2AB22 <- 164
ret <- LineCon(FitBusT2AB11, FitBusT2AB21, FitBusT2AB22, nepT2AB11, nepT2AB21, nepT2AB22)

Summary

Description

Summary of the elements of a matrix.

Usage

 SUM(A)

Arguments

Value

A list including the following components:

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

Examples

X <- matrix(rnorm(6*3),ncol=3)
summary <- SUM(X)

Interactive Tucker1 analysis

Description

Detects the underlying structure of a three-way array according to the Tucker1 (T1) model.

Usage

 
 T1(dati, laba, labb, labc)

Arguments

Value

A list including the following components:

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

References

P. Giordani, H.A.L. Kiers, M.A. Del Ferraro (2014). Three-way component analysis using the R package ThreeWay. Journal of Statistical Software 57(7):1–23. http://www.jstatsoft.org/v57/i07/.
P.M. Kroonenberg (2008). Applied Multiway Data Analysis. Wiley, New Jersey.
L.R Tucker (1966). Some mathematical notes on three-mode factor analysis. Psychometrika 31:279–311.

See Also

CP,T3,T2

Examples

data(Bus)
# labels for Bus data
laba <- rownames(Bus)
labb <- substr(colnames(Bus)[1:5],1,1)
labc <- substr(colnames(Bus)[seq(1,ncol(Bus),5)],3,8)
## Not run: 
# interactive T1 analysis
BusT1 <- T1(Bus, laba, labb, labc)
# interactive T1 analysis (when labels are not available)
BusT1 <- T1(Bus)

## End(Not run)

Tucker1 solutions

Description

Computes all the Tucker1 solutions using PCASup results with r1 (from 1 to maxa, if A-mode reduced), r2 (from 1 to maxb, if B-mode reduced) and r3 (from 1 to maxc, if C-mode reduced) components.

Usage

T1runsFit(X, n, m, p, maxa, maxb, maxc, model)

Arguments

Value

Note

Cumulative sum of eigenvalues and fits from PCAsup applied to the reduced mode are automatically printed.

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

References

H.A.L. Kiers (1991). Hierarchical relations among three-way methods. Psychometrika 56:449–470.

See Also

DimSelector, LineCon, pcasup1, T1

Examples

data(Bus)
# Fit values of T1-A with different numbers of components (from 1 to 5)
FitT1 <- T1runsFit(Bus, 7, 5, 37, 5, 5, 37, 1)

Interactive Tucker2 analysis

Description

Detects the underlying structure of a three-way array according to the Tucker2 (T2) model.

Usage

 
 T2(dati, laba, labb, labc)

Arguments

Value

A list including the following components:

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

References

P. Giordani, H.A.L. Kiers, M.A. Del Ferraro (2014). Three-way component analysis using the R package ThreeWay. Journal of Statistical Software 57(7):1–23. http://www.jstatsoft.org/v57/i07/.
P.M. Kroonenberg (2008). Applied Multiway Data Analysis. Wiley, New Jersey.
L.R Tucker (1966). Some mathematical notes on three-mode factor analysis. Psychometrika 31:279–311.

See Also

CP,T3,T1

Examples

data(Bus)
# labels for Bus data
laba <- rownames(Bus)
labb <- substr(colnames(Bus)[1:5], 1, 1)
labc <- substr(colnames(Bus)[seq(1,ncol(Bus),5)], 3, 8)
## Not run: 
# interactive T2 analysis
BusT2 <- T2(Bus, laba, labb, labc)
# interactive T2 analysis (when labels are not available)
BusT2 <- T2(Bus)

## End(Not run)

Algorithm for the Tucker2 model

Description

Alternating Least Squares algorithm for the minimization of the Tucker2 loss function.

Usage

 T2func(X, n, m, p, r1, r2, r3, start, conv, model, A, B, C, H) 
 

Arguments

Value

A list including the following components:

Note

The loss function to be minimized is ||X_A - A G_A kron(C',B')||^2 where X_A and G_A denote the matricized (frontal slices) data array and core array, respectively, and kron stands for the Kronecker product.
T2func is the same as T2funcrep except that all printings are available.

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

References

H.A.L. Kiers, P.M. Kroonenberg \& J.M.F. ten Berge (1992). An efficient algorithm for TUCKALS3 on data with large numbers of observation units. Psychometrika 57:415–422.
P.M. Kroonenberg and J. de Leeuw (1980). Principal component analysis of three-mode data by means of alternating least squares algorithms. Psychometrika 45:69–97.

See Also

T2, T2funcrep

Examples

data(Bus)
# labels for Bus data
laba <- rownames(Bus)
labb <- substr(colnames(Bus)[1:5], 1, 1)
labc <- substr(colnames(Bus)[seq(1,ncol(Bus),5)], 3, 8)
# T2-AB solution using two components for the A- and B-modes 
# (rational starting point by SVD [start=0])
BusT2 <- T2func(Bus, 7, 5, 37, 2, 2, 37, 0, 1e-6, 1)
# T2-AC solution using two components for for the A- and C-modes 
# (random orthonormalized starting point [start=1])
BusT2 <- T2func(Bus, 7, 5, 37, 2, 5, 2, 1, 1e-6, 2)
# T2-BC solution using two components for the B- and C- modes 
# (user starting point [start=2])
BusT2 <- T2func(Bus, 7, 5, 37, 7, 2, 2, 1, 1e-6, 3, diag(7), 
 matrix(rnorm(5*2),nrow=5), matrix(rnorm(37*2),nrow=37), 
 matrix(rnorm(7*4),nrow=7))

Algorithm for the Tucker2 model

Description

Alternating Least Squares algorithm for the minimization of the Tucker2 loss function.

Usage

 T2funcrep(X, n, m, p, r1, r2, r3, start, conv, model, A, B, C, H) 
 

Arguments

Value

A list including the following components:

Note

The loss function to be minimized is ||X_A - A G_A kron(C',B')||^2 where X_A and G_A denote the matricized (frontal slices) data array and core array, respectively, and kron stands for the Kronecker product.
T2funcrep is the same as T2func except that all printings are suppressed. Thus, T2funcrep can be helpful for simulation experiments.

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

References

H.A.L. Kiers, P.M. Kroonenberg \& J.M.F. ten Berge (1992). An efficient algorithm for TUCKALS3 on data with large numbers of observation units. Psychometrika 57:415–422.
P.M. Kroonenberg and J. de Leeuw (1980). Principal component analysis of three-mode data by means of alternating least squares algorithms. Psychometrika 45:69–97.

See Also

T2, T2func

Examples

data(Bus)
# labels for Bus data
laba <- rownames(Bus)
labb <- substr(colnames(Bus)[1:5], 1, 1)
labc <- substr(colnames(Bus)[seq(1,ncol(Bus),5)], 3, 8)
# T2-AB solution using two components for the A- and B-modes 
# (rational starting point by SVD [start=0])
BusT2 <- T2funcrep(Bus, 7, 5, 37, 2, 2, 37, 0, 1e-6,1)
# T2-AC solution using two components for for the A- and C-modes 
# (random orthonormalized starting point [start=1])
BusT2 <- T2funcrep(Bus, 7, 5, 37, 2, 5, 2, 1, 1e-6, 2)
# T2-BC solution using two components for the B- and C- modes 
# (user starting point [start=2])
BusT2 <- T2funcrep(Bus, 7, 5, 37, 7, 2, 2, 1, 1e-6, 3, diag(7), 
 matrix(rnorm(5*2),nrow=5), matrix(rnorm(37*2),nrow=37), 
 matrix(rnorm(7*4),nrow=7))

Approximated Tucker2 solutions

Description

Computes all the approximated Tucker2 solutions using PCASup results with r1 (from 1 to maxa, if A-mode reduced), r2 (from 1 to maxb, if B-mode reduced) and r3 (from 1 to maxc, if C-mode reduced) components.

Usage

T2runsApproxFit(X, n, m, p, maxa, maxb, maxc, model)

Arguments

Value

Note

Cumulative sum of eigenvalues and fits from PCAsup applied to the reduced modes are automatically printed.

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

References

H.A.L. Kiers (1991). Hierarchical relations among three-way methods. Psychometrika 56:449–470.

See Also

DimSelector, LineCon, pcasup2, T2

Examples

data(Bus)
# Fit values of T2-AB with different numbers of components 
# (from 1 to 3 for the B-mode, from 1 to 5 for the C-mode)
FitT2 <- T2runsApproxFit(Bus, 7, 5, 37, 7, 3, 5, 3)

Interactive Tucker3 analysis

Description

Detects the underlying structure of a three-way array according to the Tucker3 (T3) model.

Usage

 
 T3(data, laba, labb, labc)

Arguments

Value

A list including the following components:

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

References

P. Giordani, H.A.L. Kiers, M.A. Del Ferraro (2014). Three-way component analysis using the R package ThreeWay. Journal of Statistical Software 57(7):1–23. http://www.jstatsoft.org/v57/i07/.
P.M. Kroonenberg (2008). Applied Multiway Data Analysis. Wiley, New Jersey.
L.R Tucker (1966). Some mathematical notes on three-mode factor analysis. Psychometrika 31:279–311.

See Also

CP,T2,T1

Examples

data(Bus)
# labels for Bus data
laba <- rownames(Bus)
labb <- substr(colnames(Bus)[1:5],1,1)
labc <- substr(colnames(Bus)[seq(1,ncol(Bus),5)],3,8)
## Not run: 
# interactive T3 analysis
BusT3 <- T3(Bus, laba, labb, labc)
# interactive T3 analysis (when labels are not available)
BusT3 <- T3(Bus)

## End(Not run)

Plot fit of Tucker3

Description

Plots fits against numbers of dimensions, with PQR as labels and fits against number of effective paramaters.

Usage

T3dimensionalityplot(A, n, m, p)

Arguments

Note

A is usually the output of DimSelector.
The number of effective parameters in a Candecomp/Parafac analysis is discussed in Weesie and Van Houwelingen (1983).

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

References

E. Ceulemans \& H.A.L. Kiers (2006). Selecting among three-mode principal component models of different types and complexities: A numerical convex hull based method. British Journal of Mathematical and Statistical Psychology 59:133–150.
J. Weesie and H. Van Houwelingen (1983). GEPCAM users' manual (first draft). Utrecht, The Netherlands: Institute of Mathematical Statistics, State University of Utrecht.

See Also

T3, DimSelector

Examples

data(Bus)
# Fit values of T3 with different numbers of components (from 1 to 4 for the A-mode, 
# from 1 to 3 for the B-mode, from 1 to 5 for the C-mode)
FitT3 <- T3runsApproxFit(Bus,7,5,37,4,3,5)
T3dimensionalityplot(FitT3,7,5,37)

Fit of each entity per mode

Description

Computation of fit contributions by combinations of modes in case of ‘renormalization’.

Usage

T3fitpartitioning(Xprep, n, m, p, AS, BT, CU, K, renormmode, laba, labb, labc)

Arguments

Value

A list including the following components:

Note

The computation of the fit contributions by combinations of modes is done in case of ‘renormalization’.
In Tucker1, renormmode must be equal to 0.

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

See Also

T3, T2, T1

Examples

data(Bus)
# labels for Bus data
laba <- rownames(Bus)
labb <- substr(colnames(Bus)[1:5], 1, 1)
labc <- substr(colnames(Bus)[seq(1,ncol(Bus),5)], 3, 8)
# T3 solution
BusT3 <- T3funcrep(Bus, 7, 5, 37, 2, 2, 2, 0, 1e-6)
# Fitpartitioning of the T3 solution
FitT3 <- T3fitpartitioning(Bus, 7, 5, 37, BusT3$A, BusT3$B, BusT3$C, BusT3$H, 0, 
 laba, labb, labc)
# Fitpartitioning of the T3 solution (when labels are not available)
FitT3 <- T3fitpartitioning(Bus, 7, 5, 37, BusT3$A, BusT3$B, BusT3$C, BusT3$H, 0)

Algorithm for the Tucker3 model

Description

Alternating Least Squares algorithm for the minimization of the Tucker3 loss function.

Usage

 T3func(X, n, m, p, r1, r2, r3, start, conv, A, B, C, H) 
 

Arguments

Value

A list including the following components:

Note

The loss function to be minimized is ||X_A - A G_A kron(C',B')||^2 where X_A and G_A denote the matricized (frontal slices) data array and core array, respectively, and kron stands for the Kronecker product.
T3func is the same as T3funcrep except that all printings are available.

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

References

H.A.L. Kiers, P.M. Kroonenberg \& J.M.F. ten Berge (1992). An efficient algorithm for TUCKALS3 on data with large numbers of observation units. Psychometrika 57:415–422.
P.M. Kroonenberg \& J. de Leeuw (1980). Principal component analysis of three-mode data by means of alternating least squares algorithms. Psychometrika 45:69–97.

See Also

T3, T3funcrep

Examples

data(Bus)
# labels for Bus data
laba <- rownames(Bus)
labb <- substr(colnames(Bus)[1:5], 1, 1)
labc <- substr(colnames(Bus)[seq(1,ncol(Bus),5)], 3, 8)
# T3 solution using two components for all the modes 
# (rational starting point by SVD [start=0])
BusT3 <- T3func(Bus, 7, 5, 37, 2, 2, 2, 0, 1e-6)
# T3 solution using two components for all the modes 
# (random orthonormalized starting point [start=1])
BusT3 <- T3func(Bus, 7, 5, 37, 2, 2, 2, 1, 1e-6)
# T3 solution using two components for all the modes 
# (user starting point [start=2])
BusT3 <- T3func(Bus, 7, 5, 37, 2, 2, 2, 1, 1e-6, matrix(rnorm(7*2),nrow=7), 
 matrix(rnorm(5*2),nrow=5), matrix(rnorm(37*2),nrow=37), 
 matrix(rnorm(2*4),nrow=2))

Algorithm for the Tucker3 model

Description

Alternating Least Squares algorithm for the minimization of the Tucker3 loss function.

Usage

 T3funcrep(X, n, m, p, r1, r2, r3, start, conv, A, B, C, H) 
 

Arguments

Value

A list including the following components:

Note

The loss function to be minimized is ||X_A - A G_A kron(C',B')||^2 where X_A and G_A denote the matricized (frontal slices) data array and core array, respectively, and kron stands for the Kronecker product.
T3funcrep is the same as T3func except that all printings are suppressed. Thus, T3funcrep can be helpful for simulation experiments.

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

References

H.A.L. Kiers, P.M. Kroonenberg \& J.M.F. ten Berge (1992). An efficient algorithm for TUCKALS3 on data with large numbers of observation units. Psychometrika 57:415–422.
P.M. Kroonenberg \& J. de Leeuw (1980). Principal component analysis of three-mode data by means of alternating least squares algorithms. Psychometrika 45:69–97.

See Also

T3, T3func

Examples

data(Bus)
# labels for Bus data
laba <- rownames(Bus)
labb <- substr(colnames(Bus)[1:5], 1, 1)
labc <- substr(colnames(Bus)[seq(1,ncol(Bus),5)], 3, 8)
# T3 solution using two components for all the modes 
# (rational starting point by SVD [start=0])
BusT3 <- T3funcrep(Bus, 7, 5, 37, 2, 2, 2, 0, 1e-6)
# T3 solution using two components for all the modes 
# (random orthonormalized starting point [start=1])
BusT3 <- T3funcrep(Bus, 7, 5, 37, 2, 2, 2, 1, 1e-6)
# T3 solution using two components for all the modes 
# (user starting point [start=2])
BusT3 <- T3funcrep(Bus, 7, 5, 37, 2, 2, 2, 1, 1e-6, matrix(rnorm(7*2),nrow=7), 
 matrix(rnorm(5*2),nrow=5), matrix(rnorm(37*2),nrow=37), 
 matrix(rnorm(2*4),nrow=2))

Approximated Tucker3 solutions

Description

Computes all the approximated Tcker3 solutions using PCASup results with r1 (from 1 to maxa), r2 (from 1 to maxb) and r3 (from 1 to maxc) components.

Usage

T3runsApproxFit(X, n, m, p, maxa, maxb, maxc)

Arguments

Value

Note

Cumulative sum of eigenvalues and fits from PCAsup applied to the A-, B- and C-modes are automatically printed.

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

References

H.A.L. Kiers (1991). Hierarchical relations among three-way methods. Psychometrika 56:449–470.

See Also

DimSelector, LineCon, pcasup3, T3

Examples

data(Bus)
# Fit values of T3 with different numbers of components (from 1 to 4 for the A-mode, 
# from 1 to 3 for the B-mode, from 1 to 5 for the C-mode)
FitT3 <- T3runsApproxFit(Bus, 7, 5, 37, 4, 3, 5)

TV data

Description

Three-way data about ratings of 15 American television shows on 16 bipolar scales made by 30 students.

Usage

data(TV)

Format

A list containing one data.frame and three character vectors.
TV[[1]] is a data.frame with 16 rows and 450 (15 x 30) columns.
The rows refer to the American television shows.
The columns refer to the combinations of scales and students with the sclaes nested within the students.
The data.frame contains the frontal slices next to each other of the original array.
The labels for the bipolar scales are in the character vector TV[[2]].
The labels for the TV programs are in the character vector TV[[3]].
The labels for the students are in the character vector TV[[4]].

Details

The original data set consists of ratings made by 40 subjects (psychology students at the University of Western Ontario in 1981). To avoid missing data, only 30 students are considered. The ratings are made on 13-point bipolar scales. Lundy et al. (1989) perform Candecomp/Parafac on the preprocessed data. Details on preprocessing are not reported, but should be centered within TV programs and scales. Three real components are extracted. However, the unconstrained Candecomp/Parafac solution with three components suffers from the so-called degeneracy (obtained solution with highly correlated and uninterpretable dimensions). Degeneracy (see, for instance, Harshman \& Lundy, 1984; Stegeman, 2006, 2007; De Silva \& Lim, 2008; Rocci \& Giordani, 2010) can be overcome by imposing orthogonal constraints in one of the component matrices. The so-obtained solution with three components is meaningful and interpretable as described in Lundy et al. (1989).

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

References

V. De Silva \& L.-H. Lim (2008). Tensor rank and the ill-posedness of the best low-rank approximation problem. SIAM Journal on Matrix Analysis and Applications 30:1084–1127.
R.A. Harshman \& M.E. Lundy (1984). Data preprocessing and the extended PARAFAC model. In H.G. Law, C.W. Snyder Jr, J.A. Hattie, \& R.P. McDonald (Eds.): Research methods for multimode data analysis. Praeger, New York (pp. 216–284).
M.E. Lundy, R.A. Harshman \& J.B. Kruskal (1989). A two-stage procedure incorporating good features of both trilinear and quadrilinear models. In R. Coppi, S. Bolasco (Eds.): Multiway Data Analysis. Elsevier, North Holland (pp. 123–130).
R. Rocci R \& P. Giordani (2010). A weak degeneracy decomposition for the CANDECOMP/PARAFAC model. Journal of Chemometrics 24:57–66.
A. Stegeman (2006). Degeneracy in Candecomp/Parafac explained for pxpx2 arrays of rank p+1 or higher. Psychometrika 71:483–501.
A. Stegeman (2007). Degeneracy in Candecomp/Parafac and Indscal explained for several three-sliced arrays with a two-valued typical rank. Psychometrika 72:601–619.

Examples

# to perform stability check and produce bootstrap confidence intervals 
# it is useful to permute the modes so that the A-mode refers to students
data(TV)
TVdata=TV[[1]]
labSCALE=TV[[2]]
labPROGRAM=TV[[3]]
labSTUDENT=TV[[4]]
TVdata <- permnew(TVdata, 16, 15, 30)
TVdata <- permnew(TVdata, 15, 30, 16)

Bootstrap percentile intervals for CANDECOMP/PARAFAC

Description

Produces percentile intervals for all output parameters. The percentile intervals indicate the instability of the sample solutions.

Usage

 
 bootstrapCP(X, A, B, C, n, m, p, r, ort1, ort2, ort3, conv, centopt, normopt, 
  scaleopt, maxit, laba, labb, labc)

Arguments

Value

A list including the following components:

Note

The preprocessing must be done in same way as for sample analysis.
The resampling mode must be the A-mode.
The starting points for every bootstrap solution are two: rational (using SVD) and solution from the observed sample.

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

References

H.A.L. Kiers (2004). Bootstrap confidence intervals for three-way methods. Journal of Chemometrics 18:22–36.

See Also

bootstrapT3, CP, percentile95

Examples

data(TV)
TVdata=TV[[1]]
labSCALE=TV[[2]]
labPROGRAM=TV[[3]]
labSTUDENT=TV[[4]]
# permutation of the modes so that the A-mode refers to students
TVdata <- permnew(TVdata, 16, 15, 30)
TVdata <- permnew(TVdata, 15, 30, 16)
# CP solution
TVcp <- CPfuncrep(TVdata, 30, 16, 15, 2, 1, 1, 1, 0, 1e-6, 10000)
## Not run: 
# Bootstrap analysis on CP solution
boot <- bootstrapCP(TVdata, TVcp$A, TVcp$B, TVcp$C, 30, 16, 15, 2, 1, 1, 1, 
 1e-6, 0, 0, 0, 10000, labSTUDENT, labSCALE, labPROGRAM)
# Bootstrap analysis on CP solution (when labels are not available)
boot <- bootstrapCP(TVdata, TVcp$A, TVcp$B, TVcp$C, 30, 16, 15, 2, 1, 1, 1, 
 1e-6, 0, 0, 0, 10000)

## End(Not run)

Bootstrap percentile intervals for Tucker3

Description

Produces percentile intervals for all output parameters. The percentile intervals indicate the instability of the sample solutions.

Usage

 bootstrapT3(X, A, B, C, G, n, m, p, r1, r2, r3, conv, centopt, normopt, 
  optimalmatch, laba, labb, labc)

Arguments

Value

A list including the following components:

Note

The preprocessing must be done in same way as for sample analysis.
The resampling mode must be the A-mode.
The starting points for every bootstrap solution are two: rational (using SVD) and solution from the observed sample.

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

References

H.A.L. Kiers (2004). Bootstrap confidence intervals for three-way methods. Journal of Chemometrics 18:22–36.

See Also

bootstrapCP, percentile95, T3

Examples

data(Bus)
# labels for Bus data
laba <- rownames(Bus)
labb <- substr(colnames(Bus)[1:5],1,1)
labc <- substr(colnames(Bus)[seq(1,ncol(Bus),5)],3,8)
# T3 solution
BusT3 <- T3funcrep(Bus, 7, 5, 37, 2, 2, 2, 0, 1e-6)
## Not run: 
# Bootstrap analysis on T3 solution using matching via optimal transformation
boot <- bootstrapT3(Bus, BusT3$A, BusT3$B, BusT3$C, BusT3$H, 7, 5, 37, 2, 2, 2, 
 1e-6, 0, 0, 1, laba, labb, labc)
# Bootstrap analysis on T3 solution using matching via orthogonal rotation 
# (when labels are not available)
boot <- bootstrapT3(Bus, BusT3$A, BusT3$B, BusT3$C, BusT3$H, 7, 5, 37, 2, 2, 2, 
 1e-6, 0, 0, 0)

## End(Not run)

Columns concatenation

Description

Concatenates the columns of two matrices next to each other.

Usage

 ccmat(A, B)

Arguments

Value

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

Examples

X <- matrix(rnorm(6*3),ncol=3)
Y <- matrix(rnorm(6*3),ncol=3)
Z <- ccmat(X,Y)

Centering of a matricized array

Description

Centering of a matricized array across one mode (modes indicated by 1,2, or 3).

Usage

 cent3(X, n, m, p, mode)

Arguments

Value

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

References

H.A.L. Kiers (2000). Towards a standardized notation and terminology in multiway analysis. Journal of Chemometrics 14:105–122.

See Also

Cc, norm3

Examples

X <- array(c(rnorm(120)),c(6,5,4))
# matricized array
Y <- supermat(X)
# data centered across A-mode
Z <- cent3(Y$Xa, 6, 5, 4, 1)
apply(Z,2,mean)
# data centered also across B-modes (double centering)
Z <- cent3(Z, 6, 5, 4, 2)
apply(Z,1,mean)
apply(Z,2,mean)

Jointplots

Description

Program for producing jointplots in general.

Usage

 jointplotgen(K, A, B, C, fixmode, fixunit, laba, labb, labc)

Arguments

Value

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

References

P.M. Kroonenberg (2008). Applied Multiway Data Analysis. Wiley, New Jersey.

Examples

data(Bus)
# labels for Bus data
laba <- rownames(Bus)
labb <- substr(colnames(Bus)[1:5], 1, 1)
labc <- substr(colnames(Bus)[seq(1,ncol(Bus),5)], 3, 8)
# <- T3 solution
BusT3 <- T3funcrep(Bus, 7, 5, 37, 2, 2, 2, 0, 1e-6)
# Joint plot for mode C and component 2
jointplotgen(BusT3$H, BusT3$A, BusT3$B, BusT3$C, 3, 2, laba, labb, labc)

Meaudret data

Description

Three-way data about six sampling sites along a small French stream (the Meaudret) on which ten biological and chemical variables are collected four times.

Usage

data(meaudret)

Format

An array of order 6 x 10 x 4.
The A-mode entities are sampling sites (Site1, ..., Site6).
The B-mode entities are biological and chemical variables (Temp, Debi, PH, Cond, Oxyg, Biod, Chem, NH4, NO3, PO4).
The C-mode entities are months (June, August, November, February).

Details

The ranges of the variables are very different and, therefore, normalization of the raw data is recommended. The data have been used by Kiers (1991) in order to show the existing relations among three-way methods.

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

References

H.A.L. Kiers (1991). Hierarchical relations among three-way methods. Psychometrika 56:449–470.

Examples

data(meaudret)
## The labels are in the data array
laba <- dimnames(meaudret)[[1]]
labb <- dimnames(meaudret)[[2]]
labc <- dimnames(meaudret)[[3]]
## Candecomp/Parafac analysis
## Not run: 
CP(meaudret,laba,labb,labc)
## Tucker3 analysis
T3(meaudret,laba,labb,labc)
## Tucker2 analysis
T2(meaudret,laba,labb,labc)
## Tucker1 analysis
T1(meaudret,laba,labb,labc)

## End(Not run)

Normalization of a matricized array

Description

Normalization of a matricized array within one mode (modes indicated by 1,2, or 3) to sum of squares equal to product of size of other modes.

Usage

 norm3(X, n, m, p, mode)

Arguments

Value

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

References

H.A.L. Kiers (2000). Towards a standardized notation and terminology in multiway analysis. Journal of Chemometrics 14:105–122.

See Also

cent3, nrm2

Examples

X <- array(c(rnorm(120)), c(6,5,4))
# matricized array
Y <- supermat(X)
# data normalized within A-mode
Z <- norm3(Y$Xa, 6, 5, 4, 1)
apply(Z^2,1,sum)
# data normalized within C-mode
Z <- norm3(Y$Xa, 6, 5, 4, 3)
Z <- permnew(Z, 6, 5, 4)
Z <- permnew(Z, 5, 4, 6)
apply(Z^2, 1, sum)

Normalized varimax rotation

Description

Produces normalized varimax rotated version of A and rotation matrix T.

Usage

 normvari(A)

Arguments

Value

A list including the following components:

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

References

H. Kaiser (1958). The varimax criterion for analytic rotation in factor analysis. Psychometrika 23:187–200.

See Also

varim

Examples

X <- matrix(rnorm(6*3),ncol=3)
Y <- normvari(X)
# normalized varimax rotated version of X
Y$B
# rotation matrix
Y$T

Columnwise normalization of a matrix

Description

Computation of a columnwise normalized version of a matrix.

Usage

 nrm2(A)

Arguments

Value

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

See Also

Cc

Examples

X <- matrix(rnorm(6*3),ncol=3)
Y <- nrm2(X)
apply(Y^2, 2, sum)

Order

Description

In case of vectors, an ordering of its elements in ascending order is produced; in case of matrices, the ordering in ascending order refers to every column.

Usage

 ord(X)

Arguments

Value

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

Examples

# vector
x <- rnorm(6)
y <- ord(x)
# matrix
X <- matrix(rnorm(6*3),ncol=3)
Y <- ord(X)

Orthonormalization of a matrix

Description

Returns an orthonormal basis for the range of A.

Usage

 orth(A)

Arguments

Value

Note

The columns of Q span the same space as the columns of A with t(Q)Q=I.
The number of columns of Q is the rank of A.

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

Examples

X <- matrix(rnorm(6*3),ncol=3)
Y <- orth(X)

Orthomax Rotation

Description

Produces a simultaneous orthomax rotation of two matrices (using one rotation matrix).

Usage

 orthmax2(A1, A2, gam1, gam2, conv)

Arguments

Value

A list including the following components:

Note

The function to be maximized is f=sum((A1^2)-1/m1*gam1*sum((sum(A1^2))^2))^2+sum((A2^2)-1/m2*gam2*sum((sum(A2^2))^2))^2.

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

References

R. Jennrich (1970). Orthogonal rotation algorithms. Psychometrika 35:229–235.

See Also

varim

Examples

X <- matrix(rnorm(8*3),ncol=3)
Y <- matrix(rnorm(6*3),ncol=3)
orthXY <- orthmax2(X,Y,1,2)
# rotated version of X
orthXY$B1
# rotated version of Y
orthXY$B2
# rotation matrix
orthXY$T

PCA of the mean matrix

Description

Performs Principal Component Analysis (PCA) of the mean matrix aggregated over mode number indicated by aggregmode.

Usage

pcamean(X, n, m, p, laba, labb, labc)

Arguments

Value

A list including the following components:

Note

aggregmode denotes the mode over which means are computed (1 for A-mode, 2 for B-mode, 3 for C-mode).
aggregmode is provided interactively.

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

References

H. Kaiser (1958). The varimax criterion for analytic rotation in factor analysis. Psychometrika 23:187–200.
C. Harris \& H. Kaiser (1964). Some mathematical notes on three-mode factor analysis. Psychometrika 29:347–362.

Examples

data(TV)
TVdata=TV[[1]]
labSCALE=TV[[2]]
labPROGRAM=TV[[3]]
labSTUDENT=TV[[4]]
# permutation of the modes so that the A-mode refers to students
TVdata <- permnew(TVdata, 16, 15, 30)
TVdata <- permnew(TVdata, 15, 30, 16)
## Not run: 
# PCA on the mean matrix
TVpcamean <- pcamean(TVdata, 30, 16, 15, labSTUDENT, labSCALE, labPROGRAM)
# PCA on the mean matrix (when labels are not available)
TVpcamean <- pcamean(TVdata, 30, 16, 15)

## End(Not run)

PCASup Analysis

Description

Computes PCASup analysis for the direction concerning the reduced mode.

Usage

 pcasup1(X, n, m, p, model)

Arguments

Value

A list including the following components:

Note

pcasup1 computes the Tucker1 solution.
Cumulative sum of eigenvalues and fits from PCAsup applied to the reduced mode are automatically printed.

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

References

H.A.L. Kiers (1991). Hierarchical relations among three-way methods. Psychometrika 56: 449–470.
H.A.L. Kiers (2000). Towards a standardized notation and terminology in multiway analysis. Journal of Chemometrics 14:105–122.
L.R Tucker (1966). Some mathematical notes on three-mode factor analysis. Psychometrika 31: 279–311.

See Also

T1

Examples

data(Bus)
# PCA-sup for T1-B
pcasupBus <- pcasup1(Bus, 7, 5, 37, 2)

PCASup Analysis

Description

Computes PCASup analysis for the directions concerning the reduced modes.

Usage

 pcasup2(X, n, m, p, model)

Arguments

Value

A list including the following components:

Note

Cumulative sum of eigenvalues and fits from PCAsup applied to the reduced modes are automatically printed.

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

References

H.A.L. Kiers (1991). Hierarchical relations among three-way methods. Psychometrika 56: 449–470.
H.A.L. Kiers (2000). Towards a standardized notation and terminology in multiway analysis. Journal of Chemometrics 14:105–122.
L.R Tucker (1966). Some mathematical notes on three-mode factor analysis. Psychometrika 31: 279–311.

See Also

T2

Examples

data(Bus)
# PCA-sup for T2-AB
pcasupBus <- pcasup2(Bus, 7, 5, 37, 1)

PCASup Analysis

Description

Computes PCASup analysis in all the three directions.

Usage

 pcasup3(X, n, m, p)

Arguments

Value

A list including the following components:

Note

pcasup3 computes the Tucker3 solution according to Tucker (1966).
Cumulative sum of eigenvalues and fits from PCAsup applied to the A-, B- and C-modes are automatically printed.

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

References

H.A.L. Kiers (1991). Hierarchical relations among three-way methods. Psychometrika 56: 449–470.
H.A.L. Kiers (2000). Towards a standardized notation and terminology in multiway analysis. Journal of Chemometrics 14:105–122.
L.R Tucker (1966). Some mathematical notes on three-mode factor analysis. Psychometrika 31: 279–311.

See Also

T3

Examples

data(Bus)
## Not run: 
# PCA-sup
pcasupBus <- pcasup3(Bus, 7, 5, 37)

## End(Not run)

95% percentile intervals

Description

Computes 2.5% and 97.5% percentiles for all columns of X.

Usage

percentile95(X)

Arguments

Value

A list including the following components:

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

See Also

bootstrapCP,bootstrapT3

Examples

X <- matrix(rnorm(50*3),ncol=3)
perc95X <- percentile95(X)

Permutation of a matricized array

Description

Permutes the matricized (n x m x p) array X to the matricized array Y of order (m x p x n).

Usage

 permnew(X,n,m,p)

Arguments

Value

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

References

H.A.L. Kiers (2000). Towards a standardized notation and terminology in multiway analysis. Journal of Chemometrics 14:105–122.

Examples

X <- array(c(rnorm(120)),c(6,5,4))
dim(X)
# matricized array
Xa <- supermat(X)$Xa
# matricized X with the A-mode entities in its rows
dim(Xa)
# matricized X with the B-mode entities in its rows
Xb <- permnew(Xa, 6, 5, 4)
dim(Xb)
# matricized X with the C-mode entities in its rows
Xc <- permnew(Xb, 5, 4, 6)
dim(Xc)

Permutation

Description

Gives all the permutations of the first integer numbers.

Usage

 perms(n)

Arguments

Value

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

References

H.A.L. Kiers (2004). Bootstrap confidence intervals for three-way methods. Journal of Chemometrics 18:22–36.

Examples

P <- perms(4)

Phi coefficient

Description

Computes the phi coefficients among columns of two matrices.

Usage

 phi(a,b)

Arguments

Value

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

References

L.R Tucker (1951). A method for synthesis of factor analysis studies. Personnel Research Section Report No. 984. Department of the Army, Washington, DC.

Examples

X <- matrix(rnorm(6*3),ncol=3)
Y <- matrix(rnorm(6*3),ncol=3)
P <- phi(X,Y)

Array reconstruction

Description

Produces an array starting from its matricization with all the frontal slices of the array next to each other.

Usage

rarray(Xa, n, m, p)

Arguments

Value

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

References

H.A.L. Kiers (2000). Towards a standardized notation and terminology in multiway analysis. Journal of Chemometrics 14:105–122.

Examples

# matricized array (frontal slice)
Xa <- matrix(1:8,nrow=2)
X <- rarray(Xa, 2, 2, 2)
# original array
X

Scaling of the Candecomp/Parafac solution

Description

Scales the Candecomp/Parafac solution producing two component matrices normalized to unit sum of squares (and compensating this scaling in the remaining component matrix).

Usage

 renormsolCP(A, B, C, mode)

Arguments

Value

A list including the following components:

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

Examples

data(TV)
TVdata=TV[[1]]
# permutation of the modes so that the A-mode refers to students
TVdata <- permnew(TVdata, 16, 15, 30)
TVdata <- permnew(TVdata, 15, 30, 16)
# CP solution
TVcp <- CPfuncrep(TVdata, 30, 16, 15, 2, 1, 1, 1, 0, 1e-6, 10000)
# sums of squares of A, B and C
sum(TVcp$A^2)
sum(TVcp$B^2)
sum(TVcp$C^2)
# Renormalization by scaling B- and C-modes
TVcpScalBC <- renormsolCP(TVcp$A, TVcp$B, TVcp$C, 1)
# sums of squares of A, B and C after renormalization
sum(TVcpScalBC$A^2)
sum(TVcpScalBC$B^2)
sum(TVcpScalBC$C^2)

Renormalization of the Tucker3 (and Tucker2) solution

Description

Renormalizes the Tucker3 solution producing a core normalized to unit sum of squares (and compensating the core normalization in the component matrices).

Usage

 renormsolT3(A, B, C, G, mode)

Arguments

Value

A list including the following components:

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

Examples

data(Bus)
# labels for Bus data
laba <- rownames(Bus)
labb <- substr(colnames(Bus)[1:5], 1, 1)
labc <- substr(colnames(Bus)[seq(1,ncol(Bus),5)], 3, 8)
# T3 solution
BusT3 <- T3funcrep(Bus, 7, 5, 37, 2, 2, 2, 0, 1e-6)
# sums of squares of A and core
sum(BusT3$A^2)
sum(BusT3$H^2)
# Renormalization with respect to the A-mode
BusT3rA <- renormsolT3(BusT3$A, BusT3$B, BusT3$C, BusT3$H,1)
# sums of squares of A and core after renormalization
sum(BusT3rA$A^2)
sum(BusT3rA$H^2)

Split-Half Analysis

Description

Performs split-half analysis for Candecomp/Parafac.

Usage

 splithalfCP(X, n, m, p, r, centopt, normopt, scaleopt, addanal, conv, 
  maxit, ort1, ort2, ort3, laba, labb, labc)

Arguments

Value

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

References

P.M. Kroonenberg (2008). Applied Multiway Data Analysis. Wiley, New Jersey.

See Also

CP

Examples

data(TV)
TVdata=TV[[1]]
labSCALE=TV[[2]]
labPROGRAM=TV[[3]]
labSTUDENT=TV[[4]]
# permutation of the modes so that the A-mode refers to students
TVdata <- permnew(TVdata, 16, 15, 30)
TVdata <- permnew(TVdata, 15, 30, 16)
## Not run: 
# Split-half analysis on CP solution
splitCP <- splithalfCP(TVdata, 30, 16, 15, 2, 0, 0, 0, 5, 1e-6, 10000, 1, 1, 1, 
 labSTUDENT, labSCALE, labPROGRAM)
# Split-half analysis on CP solution (when labels are not available)
splitCP <- splithalfCP(TVdata, 30, 16, 15, 2, 0, 0, 0, 5, 1e-6, 10000, 1, 1, 1)

## End(Not run)

Split-Half Analysis

Description

Performs split-half analysis for Tucker3.

Usage

 splithalfT3(X, n, m, p, r1, r2, r3, centopt, normopt, renormmode, 
  wa_rel, wb_rel, wc_rel, addanal, conv, laba, labb, labc)

Arguments

Value

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

References

P.M. Kroonenberg (2008). Applied Multiway Data Analysis. Wiley, New Jersey.

See Also

T3

Examples

data(Bus)
# labels for Bus data
laba <- rownames(Bus)
labb <- substr(colnames(Bus)[1:5],1,1)
labc <- substr(colnames(Bus)[seq(1,ncol(Bus),5)],3,8)
## Not run: 
# Split-half analysis on T3 solution
splitT3 <- splithalfT3(Bus, 7, 5, 37, 2, 2, 2, 0, 0, 0, 3, 3, 0, 5, 1e-6, 
 laba, labb, labc)
# Split-half analysis on T3 solution (when labels are not available)
splitT3 <- splithalfT3(Bus, 7, 5, 37, 2, 2, 2, 0, 0, 0, 3, 3, 0, 5, 1e-6)

## End(Not run)

Matrix unfolding

Description

Produces matricizations of a three-way array into matrices denoted as super-matrices.

Usage

 supermat(X)

Arguments

Value

A list including the following components:

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

References

H.A.L. Kiers (2000). Towards a standardized notation and terminology in multiway analysis. Journal of Chemometrics 14:105–122.

Examples

# array (2x2x2) with integers from 1 to 8
X <- array(1:8,c(2,2,2))
Y <- supermat(X)
# matricized arrays
Y$Xa
Y$Xb
Y$Xc 

Three-way ANOVA

Description

Computation of three-way Analysis of Variance (ANOVA).

Usage

threewayanova(Y, n, m, p)

Arguments

Value

A list including the following components:

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

References

H.A.L. Kiers \& I. Van Mechelen (2001). Three-way component analysis: principles and illustrative applications. Psychological Methods 6:84–110.

Examples

data(TV)
TVdata=TV[[1]]
anova3 <- threewayanova(TVdata, 16, 15, 30)

Trace

Description

Computes the trace of a matrix.

Usage

 tr(a)

Arguments

Value

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

Examples

X <- matrix(rnorm(6*6),ncol=6)
trace <- tr(X)

Varimax roation

Description

Produces varimax rotated version of A and rotation matrix T.

Usage

 varim(A)

Arguments

Value

A list including the following components:

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

References

H. Kaiser (1958). The varimax criterion for analytic rotation in factor analysis. Psychometrika 23:187–200.
K. Nevels (1986). A direct solution for pairwise rotations in Kaiser's varimax method. Psychometrika 51:327–329.

See Also

normvari

Examples

X <- matrix(rnorm(6*3),ncol=3)
Y <- varim(X)
# varimax rotated version of X
Y$B
# rotation matrix
Y$T

Varimax Rotation for Tucker3 and Tucker2

Description

Performs varimax rotation of the core and component matrix rotations to simple structure.

Usage

varimcoco(A, B, C, H, wa_rel, wb_rel, wc_rel, rot1, rot2, rot3, nanal)

Arguments

Value

A list including the following components:

Note

The simplicity values f1, f2a, f2b, f2c are based on ‘natural’ weigths and therefore comparable across matrices. When multiplied by the relative weights, they give the contribution to the overall simplicity value (they are I^2/p, J^2/q or K^2/r, respectively, times the sum of the variances of squared values).

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

References

H.A.L. Kiers (1998). Joint orthomax rotation of the core and component matrices resulting from three-mode principal components analysis. Journal of Classification 15:245–263.

See Also

orthmax2, varim

Examples

data(Bus)
# T3 solution
BusT3 <- T3funcrep(Bus, 7, 5, 37, 2, 2, 2, 0, 1e-6)
# Simplicity of A (with weight = 2.5), B (with weight = 2) and C (with weight = 1.5)
T3vmABC <- varimcoco(BusT3$A, BusT3$B, BusT3$C, BusT3$H, 2.5, 2, 1.5)
# Simplicity of only A (with weight = 2.5) and B (with weight = 2)
# rot3=0; the value of wc_rel (= 0) does not play an active role
T3vmAB <- varimcoco(BusT3$A, BusT3$B, BusT3$C, BusT3$H, 2.5, 2, 0, 1, 1, 0)
# simplicity repeatedly with different relative weights for A, B and C
T3vm <- list()
weight.a <- c(1, 3, 6)
weight.b <- c(0, 2, 5)
weight.c <- c(1, 4)
i <- 1
for (wa_rel in weight.a){
 for (wb_rel in weight.b){
  for (wc_rel in weight.c){
   T3vm[[i]] <- varimcoco(BusT3$A, BusT3$B, BusT3$C, 
    BusT3$H, wa_rel, wb_rel, wc_rel)
   i <- i+1
  }
 }
}