Type:	Package
Title:	Tools for Linear Dimension Reduction
Version:	0.2-2
Date:	2023-09-17
Author:	Eero Liski [aut], Klaus Nordhausen [aut, cre], Hannu Oja [aut], Anne Ruiz-Gazen [aut]
Maintainer:	Klaus Nordhausen <klausnordhausenR@gmail.com>
Depends:	R (≥ 3.2.2)
Suggests:	dr
Description:	Linear dimension reduction subspaces can be uniquely defined using orthogonal projection matrices. This package provides tools to compute distances between such subspaces and to compute the average subspace. For details see Liski, E.Nordhausen K., Oja H., Ruiz-Gazen A. (2016) Combining Linear Dimension Reduction Subspaces <doi:10.1007/978-81-322-3643-6_7>.
License:	GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]
NeedsCompilation:	no
Packaged:	2023-09-17 18:30:57 UTC; admin
Repository:	CRAN
Date/Publication:	2023-09-17 23:12:34 UTC

Tools for Linear Dimension Reduction

Description

Linear dimension reduction subspaces can be uniquely defined using orthogonal projection matrices. This package provides tools to compute distances between such subspaces and to compute the average subspace. For details see Liski, E.Nordhausen K., Oja H., Ruiz-Gazen A. (2016) Combining Linear Dimension Reduction Subspaces <doi:10.1007/978-81-322-3643-6_7>.

Details

Index of help topics:

AOP                     Function to Average Orthogonal Projection
                        Matrices
B2P                     Function to Compute an Orthogonal Projection
                        Matrix Based on an Arbitrary Matrix
LDRTools-package        Tools for Linear Dimension Reduction
O2P                     Function to Compute an Orthogonal Projection
                        Matrix Based on a Matrix with Orthonormal
                        Columns
Pdist                   Function to Compute the Distances Between
                        Orthogonal Projection Matrices

Author(s)

Eero Liski [aut], Klaus Nordhausen [aut, cre] (<https://orcid.org/0000-0002-3758-8501>), Hannu Oja [aut] (<https://orcid.org/0000-0002-4945-5976>), Anne Ruiz-Gazen [aut] (<https://orcid.org/0000-0001-8970-8061>)

Maintainer: Klaus Nordhausen <klausnordhausenR@gmail.com>

References

Liski E., Nordhausen K., Oja H., and Ruiz-Gazen A. (2016), Combining Linear Dimension Reduction Subspaces. In: Agostinelli C., Basu A., Filzmoser P., Mukherjee D. (eds) Recent Advances in Robust Statistics: Theory and Applications. doi:10.1007/978-81-322-3643-6_7.

Function to Average Orthogonal Projection Matrices

Description

The function computes the average of orthogonal projection matrices and estimates the average rank.

Usage

AOP(x, weights = "constant")

Arguments

Details

The AOP maximizes the function D(P)= w(k)tr(\bar P_wP)- \frac 12 w^2(k)k, where \bar P_w=\frac 1m \sum_{i=1}^m w(k_i) P_i is a regular average of weighted orthogonal projection matrices, m is the number of orthogonal projection matrices averaged, w(k)is the weight function and k is the rank of P. The possible weights are defined as constant: w(k)=1, inverse: w(k)=1/k and sq.inverse: w(k)=1/\sqrt k. The constant weight corresponds to the so called Crone & Crosby distance. Orthogonal projection matrices of zero rank are also possible inputs for the function. In such a case, the function prints a warning giving the number of orthogonal projection matrices with zero rank.

Value

A list containing the following components:

Author(s)

Eero Liski and Klaus Nordhausen

References

Crone, L. J., and Crosby, D. S. (1995), Statistical Applications of a Metric on Subspaces to Satellite Meteorology, Technometrics 37, 324-328.

Liski E., Nordhausen K., Oja H., and Ruiz-Gazen A. (2016), Combining Linear Dimension Reduction Subspaces. In: Agostinelli C., Basu A., Filzmoser P., Mukherjee D. (eds) Recent Advances in Robust Statistics: Theory and Applications. doi:10.1007/978-81-322-3643-6_7.

See Also

Pdist

Examples

## Ex.1
##
library(dr)
# Australian athletes data with 202 observations
data(ais)
# 10 explanatory variables
X <- as.matrix(ais[,c(2:3,5:12)])
colnames(X) <- names(ais[,c(2:3,5:12)])
p <- dim(X)[2]
# Response variable lean body mass (LBM)
y <- ais$LBM
# Significance level 
alpha <- 0.05


# SIR
s0.sir <- dr(y ~ X, method="sir")
# Estimate of k 
k.sir <- sum(dr.test(s0.sir, numdir=4)[,3] < alpha)
# List of transformation matrices corresponding to 
# k.sir and fixed k=1, respectively
B.sir.list <- list(B1=s0.sir$evectors[,1:k.sir], B2=s0.sir$evectors[,1:1])
# List of orthogonal projectors corresponding to 
# k.sir, fixed k=1 and fixed k=0, respectively
P.sir.list <- list(P1=O2P(B.sir.list$B1), P2=O2P(B.sir.list$B2), 
P3=diag(0,p))


# SAVE
s0.save <- dr(y ~ X, method="save")
# Estimate of k 
k.save <- sum(dr.test(s0.save, numdir=4)[,3] < alpha)
# List of transformation matrices corresponding to 
# k.save and fixed k=1, respectively
B.save.list <- list(B1=s0.save$evectors[,1:k.save], 
B2=s0.save$evectors[,1:1])
# List of orthogonal projectors corresponding to 
# k.save, fixed k=1 and fixed k=0, respectively
P.save.list <- list(P1=O2P(B.save.list$B1), P2=O2P(B.save.list$B2), 
P3=diag(0,p))


# DR k-estimates
dr.k <- c(k.sir, k.save)
names(dr.k) <- c("SIR","SAVE")
dr.k


# List of individually estimated projectors
proj.list.a <- list(P.sir.list$P1, P.save.list$P1)
# List of fixed projectors
proj.list.b <- list(P.sir.list$P2, P.save.list$P2)
# List of zero projectors
proj.list.c <- list(P.sir.list$P3, P.save.list$P3)
# List of zero-rank SIR-projector and 
# other individually estimated projectors
proj.list.d <- list(P.sir.list$P3, P.save.list$P1)


# AOP (constant) object corresponding to the first projector list
AOP.const.a <- AOP(proj.list.a, weights="constant")

# AOP (inverse) objects corresponding to three projector lists
AOP.inv.a <- AOP(proj.list.a, weights="inverse")
AOP.inv.b <- AOP(proj.list.b, weights="inverse")
AOP.inv.c <- AOP(proj.list.c, weights="inverse")

# AOP (sq.inverse) objects corresponding to three projector lists
AOP.sqinv.a <- AOP(proj.list.a, weights="sq.inverse")
AOP.sqinv.c <- AOP(proj.list.c, weights="sq.inverse")
AOP.sqinv.d <- AOP(proj.list.d, weights="sq.inverse")


# k-estimates of the AOP's
AOP.a <- c(AOP.const.a$k, AOP.inv.a$k, AOP.sqinv.a$k)
names(AOP.a) <- c("const","inv","sqinv")
AOP.a

AOP.c <- AOP.inv.c$k
names(AOP.c) <- c("inv")
AOP.c

AOP.d <- AOP.sqinv.d$k
names(AOP.d) <- c("sqinv")
AOP.d


# Scatter plots between the response and the transformed data 
# corresponding to the different AOP transformation matrices

# AOP.inverse
newdata.inv.AOPa <- cbind(y,X %*% AOP.inv.a$O)
pairs(newdata.inv.AOPa)

newdata.inv.AOPb <- cbind(y,X %*% AOP.inv.b$O)
pairs(newdata.inv.AOPb)


# AOP.sq.inverse
newdata.sqinv.AOPc <- cbind(y,X %*% AOP.sqinv.c$O)
pairs(newdata.sqinv.AOPc)

newdata.sqinv.AOPd <- cbind(y,X %*% AOP.sqinv.d$O)
pairs(newdata.sqinv.AOPd)






###################################
## Ex.2
##
a <- c(1,1,rep(0,8))
A <- diag(a)
B <- diag(0,10)
B[3,1] <- 1
P.A <- O2P(A[,1:2])
P.B <- O2P(B[,1])
zero.mat <- diag(0,10)
# True projector, k=3
P.C <- P.A + P.B

# Average P.A and P.B
proj.list <- list(P.A, P.B)
AOP.const <- AOP(proj.list, weights="constant")
AOP.inv <- AOP(proj.list, weights="inverse")
AOP.sqinv <- AOP(proj.list, weights="sq.inverse")
k.list <- c(AOP.const$k, AOP.inv$k, AOP.sqinv$k)
names(k.list) <- c("const","inv","sqinv")
k.list

# Average P.A, P.B and three zero rank matrices
proj.list <- list(P.A, P.B, zero.mat, zero.mat, zero.mat)
AOP.const <- AOP(proj.list, weights="constant")
AOP.inv <- AOP(proj.list, weights="inverse")
AOP.sqinv <- AOP(proj.list, weights="sq.inverse")
k.list <- c(AOP.const$k, AOP.inv$k, AOP.sqinv$k)
names(k.list) <- c("const","inv","sqinv")
k.list

Function to Compute an Orthogonal Projection Matrix Based on an Arbitrary Matrix

Description

Function to compute an orthogonal projection matrix based on an arbitrary matrix.

Usage

B2P(x)

Arguments

Details

The orthogonal projection matrix P corresponding to matrix x is defined as P=x(x^{T}x)^{-1}x^{T}.

Value

The resulting orthogonal projection matrix.

Author(s)

Klaus Nordhausen

See Also

O2P

Examples

set.seed(1)
X <- matrix(rnorm(30),ncol=3)
P <- B2P(X)

Function to Compute an Orthogonal Projection Matrix Based on a Matrix with Orthonormal Columns

Description

Function to compute an orthogonal projection matrix based on a matrix with orthonormal columns.

Usage

O2P(x)

Arguments

Details

The orthogonal projection matrix P corresponding to matrix x is defined as P=xx^{T}.

Value

The resulting orthogonal projection matrix.

Author(s)

Klaus Nordhausen

See Also

B2P

Examples

X <- tcrossprod(matrix(rnorm(100),ncol=10))
# Orthogonal projector based on the first three eigenvectors of X
P <- O2P(eigen(X)$vectors[,1:3])

Function to Compute the Distances Between Orthogonal Projection Matrices

Description

The function computes distances between orthogonal projection matrices that might have different ranks. Different weight functions for the ranks are available.

Usage

Pdist(x, weights = "constant")

Arguments

Details

A weighted distance between subspaces P_1 and P_2 with ranks k_1 and k_2 is given by D_{w}^2(P_1,P_2)=\frac{1}{2} ||w(k_1)P_1-w(k_2)P_2||^2, where w denotes the weight function. The possible weights are defined as constant: w(k)=1, inverse: w(k)=1/k and sq.inverse: w(k)=1/\sqrt k. The constant weight corresponds to the so called Crone & Crosby distance. Orthogonal projection matrices of zero rank are also possible inputs for the function.

Value

an object of class dist having the attributes:

Author(s)

Eero Liski and Klaus Nordhausen

References

Crone, L. J., and Crosby, D. S. (1995), Statistical Applications of a Metric on Subspaces to Satellite Meteorology, Technometrics 37, 324-328.

Liski E., Nordhausen K., Oja H., and Ruiz-Gazen A. (2016), Combining Linear Dimension Reduction Subspaces. In: Agostinelli C., Basu A., Filzmoser P., Mukherjee D. (eds) Recent Advances in Robust Statistics: Theory and Applications. doi:10.1007/978-81-322-3643-6_7.

See Also

AOP

Examples

# Ex.1
X.1 <- tcrossprod(matrix(rnorm(16),ncol=4))
X.2 <- tcrossprod(matrix(rnorm(16),ncol=4))
X.3 <- tcrossprod(matrix(rnorm(16),ncol=4))
U1 <- eigen(X.1)$vectors
U2 <- eigen(X.2)$vectors
U3 <- eigen(X.3)$vectors

PRO <- list(P1=O2P(U1),P2=O2P(U2),P3=O2P(U3))

DIST.MAT<-Pdist(PRO)
str(DIST.MAT)
as.matrix(DIST.MAT)
print(DIST.MAT, diag=TRUE)
print(DIST.MAT, diag=TRUE, upper=TRUE)

PRO2 <- list(O2P(U1),O2P(U2),O2P(U3))
Pdist(PRO2, weights="inverse")

#############################
# Ex.2
a <- c(1,1,rep(0,8))
A <- diag(a)
b <- c(1,1,1,1,rep(0,6))
B <- diag(b)
P.A <- O2P(A[,1:2])
P.B <- O2P(B[,1:4])

proj.list <- list(P.A,P.B)
Pdist(proj.list, weights="constant")
Pdist(proj.list, weights="inverse")
Pdist(proj.list, weights="sq.inverse")