Package 'cml'

Conditional Manifold Learning

Description

Finds a low-dimensional embedding of high-dimensional data, conditioning on available manifold information. The current version supports conditional MDS (based on either conditional SMACOF or closed-form solution) and conditional ISOMAP.

Please cite this package as follows:

Bui, A.T. (2021). Dimension Reduction with Prior Information for Knowledge Discovery. arXiv:2111.13646. https://arxiv.org/abs/2111.13646

Bui, A. T. (2022). A Closed-Form Solution for Conditional Multidimensional Scaling. Pattern Recognition Letters 164, 148-152. https://doi.org/10.1016/j.patrec.2022.11.007

Details

Brief descriptions of the main functions of the package are provided below:

condMDS(): is the conditional MDS method, which uses conditional SMACOF to optimize its conditional stress objective function.

condMDSeigen(): is the conditional MDS method, which uses a closed-form solution based on multiple linear regression and eigendecomposition.

condIsomap(): is the conditional ISOMAP method, which is basically conditional MDS applying to graph distances (i.e., estimated geodesic distances) of the given distances/dissimilarities.

Author(s)

Anh Tuan Bui

Maintainer: Anh Tuan Bui <[email protected]>

References

Bui, A.T. (2021). Dimension Reduction with Prior Information for Knowledge Discovery. arXiv:2111.13646. https://arxiv.org/abs/2111.13646.

Bui, A. T. (2022). A Closed-Form Solution for Conditional Multidimensional Scaling. Pattern Recognition Letters 164, 148-152. https://doi.org/10.1016/j.patrec.2022.11.007

Examples

## Generate car-brand perception data
factor.weights <- c(90, 88, 83, 82, 81, 70, 68)/562
N <- 100
set.seed(1)
data <- matrix(runif(N*7), N, 7)
colnames(data) <- c('Quality', 'Safety', 'Value',	'Performance', 'Eco', 'Design', 'Tech')
rownames(data) <- paste('Brand', 1:N)
data.hat <- data + matrix(rnorm(N*7), N, 7)*data*.05
data.weighted <- t(apply(data, 1, function(x) x*factor.weights))
d <- dist(data.weighted)
d.hat <- d + rnorm(length(d))*d*.05

## The following examples use the first 4 factors as known features
# Conditional MDS based on conditional SMACOF
u.cmds = condMDS(d.hat, data.hat[,1:4], 3, init='none')
u.cmds$B # compare with diag(factor.weights[1:4])
ccor(data.hat[,5:7], u.cmds$U)$cancor # canonical correlations
vegan::procrustes(data.hat[,5:7], u.cmds$U, symmetric = TRUE)$ss # Procrustes statistic

# Conditional MDS based on the closed-form solution
u.cmds = condMDSeigen(d.hat, data.hat[,1:4], 3)
u.cmds$B # compare with diag(factor.weights[1:4])
ccor(data.hat[,5:7], u.cmds$U)$cancor # canonical correlations
vegan::procrustes(data.hat[,5:7], u.cmds$U, symmetric = TRUE)$ss # Procrustes statistic

# Conditional MDS based on conditional SMACOF,
# initialized by the closed-form solution
u.cmds = condMDS(d.hat, data.hat[,1:4], 3, init='eigen')
u.cmds$B # compare with diag(factor.weights[1:4])
ccor(data.hat[,5:7], u.cmds$U)$cancor # canonical correlations
vegan::procrustes(data.hat[,5:7], u.cmds$U, symmetric = TRUE)$ss # Procrustes statistic

# Conditional ISOMAP
u.cisomap = condIsomap(d.hat, data.hat[,1:4], 3, k = 20, init='eigen')
u.cisomap$B # compare with diag(factor.weights[1:4])
ccor(data.hat[,5:7], u.cisomap$U)$cancor
vegan::procrustes(data.hat[,5:7], u.cisomap$U, symmetric = TRUE)$ss

---

Canonical Correlations

Description

Computes canonical correlations for two sets of multivariate data x and y.

Usage

ccor(x, y)

Arguments

x	the first multivariate dataset.
y	the second multivariate dataset.

Value

a list of the following components:

cancor	a vector of canonical correlations.
xcoef	a matrix, each column of which is the vector of coefficients of x to produce the corresponding canonical covariate.
ycoef	a matrix, each column of which is the vector of coefficients of y to produce the corresponding canonical covariate.

Author(s)

Anh Tuan Bui

Examples

ccor(iris[,1:2], iris[,3:4])

---

Conditional Euclidean distance

Description

Internal functions.

Usage

condDist(U, V.tilda, one_n_t=t(rep(1,nrow(U))))
condDist2(U, V.tilda2, one_n_t=t(rep(1,nrow(U))))

Arguments

U	the embedding U
V.tilda	= V %*% B
V.tilda2	= V %*% b^2*t(V)
one_n_t	= t(rep(1,nrow(U)))

Value

a dist object.

Author(s)

Anh Tuan Bui

References

Bui, A.T. (2021). Dimension Reduction with Prior Information for Knowledge Discovery. arXiv:2111.13646. https://arxiv.org/abs/2111.13646.

---

Conditional ISOMAP

Description

Finds a low-dimensional manifold embedding of a given distance/dissimilarity matrix, conditioning on available manifold information. The method applies conditional MDS (see condMDS) to a graph distance matrix computed for the given distances/dissimilarities, using the isomap{vegan} function.

Usage

condIsomap(d, V, u.dim, epsilon = NULL, k, W,
           method = c('matrix', 'vector'), exact = TRUE,
           it.max = 1000, gamma = 1e-05,
           init = c('none', 'eigen', 'user'),
           U.start, B.start, ...)

Arguments

d	a distance/dissimilarity matrix of N entities (or a dist object).
V	an Nxq matrix of q manifold auxiliary parameter values of the N entities.
u.dim	the embedding dimension.
epsilon	shortest dissimilarity retained.
k	Number of shortest dissimilarities retained for a point. If both epsilon and k are given, epsilon will be used.
W	an NxN symmetric weight matrix. If not given, a matrix of ones will be used.
method	if matrix, there are no restrictions for the B matrix . If vector, the B matrix is restricted to be diagonal. The latter is more efficient for large q.
exact	only relevant if W is not given. In this case, if exact == FALSE, U is updated by the large-N approximation formula.
it.max	the max number of conditional SMACOF iterations.
gamma	conditional SMACOF stops early if the reduction of normalized conditional stress is less than gamma
init	initialization method.
U.start	user-defined starting values for the embedding (when init = 'user')
B.start	starting B matrix.
...	other arguments for the isomap{vegan} function.

Value

U	the embedding result.
B	the estimated B matrix.
stress	Normalized conditional stress value.
sigma	the conditional stress value at each iteration.
init	the value of the init argument.
U.start	the starting values for the embedding.
B.start	starting values for the B matrix.
method	the value of the method argument.
exact	the value of the exact argument.

Author(s)

Anh Tuan Bui

References

Bui, A.T. (2021). Dimension Reduction with Prior Information for Knowledge Discovery. arXiv:2111.13646. https://arxiv.org/abs/2111.13646.

Bui, A. T. (2022). A Closed-Form Solution for Conditional Multidimensional Scaling. Pattern Recognition Letters 164, 148-152. https://doi.org/10.1016/j.patrec.2022.11.007

See Also

condMDS, condMDSeigen

Examples

# see help(cml)

---

Conditional Multidimensional Scaling

Description

Wrapper of condSmacof, which finds a low-dimensional embedding of a given distance/dissimilarity matrix, conditioning on available manifold information.

Usage

condMDS(d, V, u.dim, W,
        method = c('matrix', 'vector'), exact = TRUE,
        it.max = 1000, gamma = 1e-05,
        init = c('none', 'eigen', 'user'),
        U.start, B.start)

Arguments

d	a distance/dissimilarity matrix of N entities (or a dist object).
V	an Nxq matrix of q manifold auxiliary parameter values of the N entities.
u.dim	the embedding dimension.
W	an NxN symmetric weight matrix. If not given, a matrix of ones will be used.
method	if matrix, there are no restrictions for the B matrix . If vector, the B matrix is restricted to be diagonal. The latter is more efficient for large q.
exact	only relevant if W is not given. In this case, if exact == FALSE, U is updated by the large-N approximation formula.
it.max	the max number of conditional SMACOF iterations.
gamma	conditional SMACOF stops early if the reduction of normalized conditional stress is less than gamma
init	initialization method.
U.start	user-defined starting values for the embedding (when init = 'user')
B.start	starting B matrix.

Value

U	the embedding result.
B	the estimated B matrix.
stress	Normalized conditional stress value.
sigma	the conditional stress value at each iteration.
init	the value of the init argument.
U.start	the starting values for the embedding.
B.start	starting values for the B matrix.
method	the value of the method argument.
exact	the value of the exact argument.

Author(s)

Anh Tuan Bui

References

Bui, A.T. (2021). Dimension Reduction with Prior Information for Knowledge Discovery. arXiv:2111.13646. https://arxiv.org/abs/2111.13646.

Bui, A. T. (2022). A Closed-Form Solution for Conditional Multidimensional Scaling. Pattern Recognition Letters 164, 148-152. https://doi.org/10.1016/j.patrec.2022.11.007

See Also

condSmacof, condMDSeigen, condIsomap

Examples

# see help(cml)

---

Conditional Multidimensional Scaling With Closed-Form Solution

Description

Provides a closed-form solution for conditional multidimensional scaling, based on multiple linear regression and eigendecomposition.

Usage

condMDSeigen(d, V, u.dim, method = c('matrix', 'vector'))

Arguments

d	a dist object of N entities.
V	an Nxq matrix of q manifold auxiliary parameter values of the N entities.
u.dim	the embedding dimension.
method	if matrix, there are no restrictions for the B matrix . If vector, the B matrix is restricted to be diagonal.

Value

U	the embedding result.
B	the estimated B matrix.
eig	the computed eigenvalues.
stress	the corresponding normalized conditional stress value of the solution.

Author(s)

Anh Tuan Bui

References

Bui, A. T. (2022). A Closed-Form Solution for Conditional Multidimensional Scaling. Pattern Recognition Letters 164, 148-152. https://doi.org/10.1016/j.patrec.2022.11.007

See Also

condMDS, condIsomap

Examples

# see help(cml)

---

Conditional SMACOF

Description

Conditional SMACOF algorithms. Intended for internal usage.

Usage

condSmacof(d, V, u.dim, W,
           method = c('matrix', 'vector'), exact = TRUE,
           it.max = 1000, gamma = 1e-05,
           init = c('none', 'eigen', 'user'),
           U.start, B.start)

Arguments

d	a dist object of N entities.
V	an Nxq matrix of q manifold auxiliary parameter values of the N entities.
u.dim	the embedding dimension.
W	an NxN symmetric weight matrix. If not given, a matrix of ones will be used.
method	if matrix, there are no restrictions for the B matrix . If vector, the B matrix is restricted to be diagonal. The latter is more efficient for large q.
exact	only relevant if W is not given. In this case, if exact == FALSE, U is updated by the large-N approximation formula.
it.max	the max number of conditional SMACOF iterations.
gamma	conditional SMACOF stops early if the reduction of normalized conditional stress is less than gamma
init	initialization method.
U.start	user-defined starting values for the embedding (when init = 'user')
B.start	starting B matrix.

Value

U	the embedding result.
B	the estimated B matrix.
stress	Normalized conditional stress value.
sigma	the conditional stress value at each iteration.
init	the value of the init argument.
U.start	the starting values for the embedding.
B.start	starting values for the B matrix.
method	the value of the method argument.
exact	the value of the exact argument.

Author(s)

Anh Tuan Bui

References

Bui, A.T. (2021). Dimension Reduction with Prior Information for Knowledge Discovery. arXiv:2111.13646. https://arxiv.org/abs/2111.13646.

Bui, A. T. (2022). A Closed-Form Solution for Conditional Multidimensional Scaling. Pattern Recognition Letters. https://doi.org/10.1016/j.patrec.2022.11.007

---

C(Z)

Description

Internal function.

Usage

cz(w, d, dz)

Arguments

w	the dist object of a weight matrix.
d	the dist object of a distance/dissimilarity matrix.
dz	the dist object of conditional distances.

Value

the matrix C(Z)

Author(s)

Anh Tuan Bui

References

Bui, A.T. (2021). Dimension Reduction with Prior Information for Knowledge Discovery. arXiv:2111.13646. https://arxiv.org/abs/2111.13646.

---

Moore-Penrose Inverse

Description

Computes the Moore-Penrose inverse (a.k.a., generalized inverse or pseudoinverse) of a matrix based on singular-value decomposition (SVD).

Usage

mpinv(A, eps = sqrt(.Machine$double.eps))

Arguments

A	a matrix of real numbers.
eps	a threshold (to be multiplied with the largest singular value) for dropping SVD parts that correspond to small singular values.

Value

the Moore-Penrose inverse.

Author(s)

Anh Tuan Bui

Examples

mpinv(2*diag(4))

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