Version: 1.3.3
Date: 2020-12-15
Title: Model-Based Sliced Inverse Regression
Description: An R package for dimension reduction based on finite Gaussian mixture modeling of inverse regression.
Depends: R (≥ 3.0)
Imports: mclust (≥ 5.4), stats, utils, graphics, grDevices
Suggests: knitr (≥ 1.20), rmarkdown (≥ 1.10), rgl (≥ 0.100)
License: GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]
VignetteBuilder: knitr
URL: https://mclust-org.github.io/msir/
Repository: CRAN
ByteCompile: true
LazyData: yes
Encoding: UTF-8
NeedsCompilation: no
Packaged: 2020-12-16 09:45:43 UTC; luca
Author: Luca Scrucca ORCID iD [aut, cre]
Maintainer: Luca Scrucca <luca.scrucca@unipg.it>
Date/Publication: 2020-12-16 12:20:02 UTC

Model-based Sliced Inverse Regression (MSIR)

Description

An R package that implements MSIR, a dimension reduction method based on Gaussian finite mixture models. The basis of the subspace is estimated by modeling the inverse distribution within slice using finite mixtures of Gaussians, with number of components and covariance matrix parameterization selected by BIC or defined by the user. The method provides an extension to sliced inverse regression (SIR) and allows to overcome the main limitation of SIR, i.e., the failure in the presence of regression symmetric relationships, without the need to impose further assumptions.

Author(s)

Luca Scrucca luca.scrucca@unipg.it

References

Scrucca, L. (2011) Model-based SIR for dimension reduction. Computational Statistics & Data Analysis, 55(11), 3010-3026.

See Also

msir

Local Polynomial Regression Fitting with Variability bands

Description

Nonparametric estimation of mean function with variability bands.

Usage

loess.sd(x, y = NULL, nsigma = 1, ...)

panel.loess(x, y, col = par("col"), bg = NA, pch = par("pch"), cex = 1, 
            col.smooth = "red", span = 2/3, degree = 2, nsigma = 1, ...)

Arguments

x

a vector of values for the predictor variable x.

y

a vector of values for the response variable y.

nsigma

a multiplier for the standard deviation function.

col, bg, pch, cex

numeric or character codes for the color(s), point type and size of points; see also par.

col.smooth

color to be used by lines for drawing the smooths.

span

smoothing parameter for loess.

degree

the degree of the polynomials to be used, see loess.

...

further argument passed to the function loess.

Value

The function loess.sd computes the loess smooth for the mean function and the mean plus and minus k times the standard deviation function.

The function panel.loess can be used to add to a scatterplot matrix panel a smoothing of mean function using loess with variability bands at plus and minus nsigmas times the standard deviation.

Author(s)

Luca Scrucca luca.scrucca@unipg.it

References

Weisberg, S. (2005) Applied Linear Regression, 3rd ed., Wiley, New York, pp. 275-278.

See Also

loess

Examples

data(cars)
plot(cars, main = "lowess.sd(cars)")
lines(l <- loess.sd(cars))
lines(l$x, l$upper, lty=2)
lines(l$x, l$lower, lty=2)

Model-based Sliced Inverse Regression (MSIR)

Description

A dimension reduction method based on Gaussian finite mixture models which provides an extension to sliced inverse regression (SIR). The basis of the subspace is estimated by modeling the inverse distribution within slice using Gaussian finite mixtures with number of components and covariance matrix parameterization selected by BIC or defined by the user.

Usage

msir(x, y, nslices = msir.nslices, slice.function = msir.slices, 
     modelNames = NULL, G = NULL, cov = c("mle", "regularized"), ...)

Arguments

x

A (n \times p) design matrix containing the predictors data values.

y

A (n \times 1) vector of data values for the response variable. It can be a numeric vector (regression) but also a factor (classification). In the latter case, the levels of the factor define the slices used.

nslices

The number of slices used, unless y is a factor. By default the value returned by msir.nslices.

slice.function

The slice functions to be used, by default msir.slices, but the user can provide a different slicing function.

modelNames

A vector of character strings indicating the Gaussian mixture models to be fitted as described in mclustModelNames. If a vector of strings is given they are used for all the slices. If a list of vectors is provided then each vector refers to a single slice.

G

An integer vector specifying the numbers of mixture components used in fitting Gaussian mixture models. If a list of vectors is provided then each vector refers to a single slice.

cov

The predictors marginal covariance matrix. Possible choices are:

  • "mle": for the maximum likelihood estimate

  • "regularized": for a regularized estimate of the covariance matrix (see msir.regularizedSigma)

  • R matrix: a (p \times p) user defined covariance matrix

...

other arguments passed to msir.compute.

Value

Returns an object of class 'msir' with attributes:

call

the function call.

x

the design matrix.

y

the response vector.

slice.info

output from slicing function.

mixmod

a list of finite mixture model objects as described in mclustModel.

loglik

the log-likelihood for the mixture models.

f

a vector of length equal to the total number of mixture components containing the fraction of observations in each fitted component within slices.

mu

a matrix of component within slices predictors means.

sigma

the marginal predictors covariance matrix.

M

the msir kernel matrix.

evalues

the eigenvalues from the generalized eigen-decomposition of M.

evectors

the raw eigenvectors from the generalized eigen-decomposition of M ordered according to the eigenvalues.

basis

the normalized eigenvectors from the generalized eigen-decomposition of M ordered according to the eigenvalues.

std.basis

standardized basis vectors obtained by multiplying each coefficient of the eigenvectors by the standard deviation of the corresponding predictor. The resulting coefficients are scaled such that all predictors have unit standard deviation.

numdir

the maximal number of directions estimated.

dir

the estimated MSIR directions from mean-centered predictors.

Author(s)

Luca Scrucca luca.scrucca@unipg.it

References

Scrucca, L. (2011) Model-based SIR for dimension reduction. Computational Statistics & Data Analysis, 55(11), 3010-3026.

See Also

summary.msir, plot.msir.

Examples

# 1-dimensional simple regression
n <- 200
p <- 5
b <- as.matrix(c(1,-1,rep(0,p-2)))
x <- matrix(rnorm(n*p), nrow = n, ncol = p)
y <- exp(0.5 * x%*%b) + 0.1*rnorm(n)
MSIR <- msir(x, y)
summary(MSIR)
plot(MSIR, type = "2Dplot")

# 1-dimensional symmetric response curve
n <- 200
p <- 5
b <- as.matrix(c(1,-1,rep(0,p-2)))
x <- matrix(rnorm(n*p), nrow = n, ncol = p)
y <- (0.5 * x%*%b)^2 + 0.1*rnorm(n)
MSIR <- msir(x, y)
summary(MSIR)
plot(MSIR, type = "2Dplot")
plot(MSIR, type = "coefficients")

# 2-dimensional response curve
n <- 300
p <- 5
b1 <- c(1, 1, 1, rep(0, p-3))
b2 <- c(1,-1,-1, rep(0, p-3))
b <- cbind(b1,b2)
x <- matrix(rnorm(n*p), nrow = n, ncol = p)
y <- x %*% b1 + (x %*% b1)^3 + 4*(x %*% b2)^2 + rnorm(n)
MSIR <- msir(x, y)
summary(MSIR)
plot(MSIR, which = 1:2)
## Not run: plot(MSIR, type = "spinplot")
plot(MSIR, which = 1, type = "2Dplot", span = 0.7)
plot(MSIR, which = 2, type = "2Dplot", span = 0.7)

Internal 'msir' functions

Description

Internal functions for package 'msir'.

Usage

msir.fit(data, labels, G = NULL, modelNames = NULL, 
         control = emControl(itmax = c(.Machine$integer.max, 50)), 
         initialization = NULL, warn = FALSE, verbose = FALSE, ...) 

eigen.decomp(X1, X2, inv = FALSE, tol = sqrt(.Machine$double.eps))

mvdnorm(x, mean, sigma, log = FALSE, tol = sqrt(.Machine$double.eps))

normalize(x)
    
msir.parameters(object, numdir = object$numdir)

msir.components(object)
msir.componentsSlice(object)

msir.recoverdir(object, data, normalized = TRUE, std = FALSE)

Details

These functions are not intended to be called by the user.

Author(s)

Luca Scrucca luca.scrucca@unipg.it

See Also

msir

BIC-type criterion for dimensionality

Description

BIC-type criterion for selecting the dimensionality of a dimension reduction subspace.

Usage

msir.bic(object, type = 1, plot = FALSE)

bicDimRed(M, x, nslices, type = 1, tol = sqrt(.Machine$double.eps))

Arguments

object

a 'msir' object

plot

if TRUE a plot of the criterion is shown.

M

the kernel matrix. See details below.

x

the predictors data matrix. See details below.

type

See details below.

nslices

the number of slices. See details below.

tol

a tolerance value

Details

This BIC-type criterion for the determination of the structural dimension selects d as the maximizer of

G(d) = l(d) - Penalty(p,d,n)

where l(d) is the log-likelihood for dimensions up to d, p is the number of predictors, and n is the sample size. The term Penalty(p,d,n) is the type of penalty to be used:

Value

Returns a list with components:

evalues

eigenvalues

l

log-likelihood

crit

BIC-type criterion

d

selected dimensionality

The msir.bic also assign the above information to the corresponding 'msir' object.

Author(s)

Luca Scrucca luca.scrucca@unipg.it

References

Zhu, Miao and Peng (2006) "Sliced Inverse Regression for CDR Space Estimation", JASA.
Zhu, Zhu (2007) "On kernel method for SAVE", Journal of Multivariate Analysis.

See Also

msir

Examples

# 1-dimensional symmetric response curve
n <- 200
p <- 5
b <- as.matrix(c(1,-1,rep(0,p-2)))
x <- matrix(rnorm(n*p), nrow = n, ncol = p)
y <- (0.5 * x%*%b)^2 + 0.1*rnorm(n)
MSIR <- msir(x, y)
msir.bic(MSIR, plot = TRUE)
summary(MSIR)
msir.bic(MSIR, type = 3, plot = TRUE)
summary(MSIR)

Default number of slices

Description

This function computes a Sturges' type number of slices to be used as default in the msir function.

Usage

msir.nslices(n, p)

Arguments

n

the number of observations in the sample.

p

the number of predictors in the sample.

Value

The function returns a single value, i.e. the number of slices.

Author(s)

Luca Scrucca luca.scrucca@unipg.it

See Also

msir

Permutation test for dimensionality

Description

Approximates marginal dimension test significance levels by sampling from the permutation distribution.

Usage

msir.permutation.test(object, npermute = 99, numdir = object$numdir, verbose = TRUE)

Arguments

object

a 'msir' object.

npermute

number of permutations to compute.

numdir

maximum value of the dimension to test.

verbose

if TRUE a textual progress bar is shown during computation.

Details

The function approximates significance levels of the marginal dimension tests based on a permutation test.

Value

The function returns a list with components:

summary

a table containing the hypotheses, the test statistics, the permutation p-values.

npermute

the number of permutations used.

Furthermore, it also assigns the above information to the corresponding 'msir' object.

Author(s)

Luca Scrucca luca.scrucca@unipg.it

References

Scrucca, L. (2011) Model-based SIR for dimension reduction. Computational Statistics & Data Analysis, 55(11), 3010-3026.

See Also

Function dr() in package dr.

Examples

## Not run: 
# 1-dimensional simple regression
n <- 200
p <- 5
b <- as.matrix(c(1,-1,rep(0,p-2)))
x <- matrix(rnorm(n*p), nrow = n, ncol = p)
y <- exp(0.5 * x%*%b) + 0.1*rnorm(n)
MSIR <- msir(x, y)
msir.permutation.test(MSIR)
summary(MSIR)

## End(Not run)

Regularized estimate of predictors covariance matrix.

Description

This function computes a regularized version of the covariance matrix of the predictors. Among the possible models the one which maximizes BIC is returned.

Usage

msir.regularizedSigma(x, inv = FALSE, model = c("XII", "XXI", "XXX"))

Arguments

x

Ahe predictors data matrix.

inv

A logical specifying what must be returned. If TRUE the inverse of the estimated covariance matrix is returned, otherwise the estimated covariance matrix (default).

model

A character string specifying the available models:

  • XII: diagonal equal variances

  • XXI: diagonal unequal variances

  • XXX: full covariance matrix

Value

A (p \times p) covariance matrix estimate.

Author(s)

Luca Scrucca luca.scrucca@unipg.it

See Also

msir

Slice a vector into slices of approximately equal size

Description

Function used for slicing a continuous response variable.

Usage

msir.slices(y, nslices)

Arguments

y

a vector of n values

nslices

the number of slices, no larger than n

Value

Returns a list with components:

slice.indicator

an indicator variable for the slices.

nslices

the actual number of slices produced.

slice.sizes

the number of observations in each slice.

Author(s)

Luca Scrucca luca.scrucca@unipg.it

See Also

msir

Plot method for 'msir' objects.

Description

Plots directions and other information from MSIR estimation.

Usage

## S3 method for class 'msir'
plot(x, which, 
     type = c("pairs", "2Dplot", "spinplot", "evalues", "coefficients"), 
     span = NULL, std = TRUE, ylab, xlab, restore.par = TRUE, ...)

Arguments

x

a 'msir' object.

which

a vector of value(s) giving the directions for which the plot should be drawn.

type

the type of plot to be drawn.

span

the span of smoother (only for type = "pairs" | "2Dplot").

std

if TRUE coefficients are standardized (only for type = "coefficients").

ylab

a character string for the y-axis label.

xlab

a character string for the x-axis label.

restore.par

if TRUE the graphical parameters (see par) changed are restored to the previous state. If you want to manipulate the resulting plot you should set restore.par = FALSE.

...

additional arguments.

Author(s)

Luca Scrucca luca.scrucca@unipg.it

References

Scrucca, L. (2011) Model-based SIR for dimension reduction. Computational Statistics & Data Analysis, 55(11), 3010-3026.

See Also

msir

Examples

## Not run: 
# 2-dimensional response curve
n <- 300
p <- 5
b1 <- c(1, 1, 1, rep(0, p-3))
b2 <- c(1,-1,-1, rep(0, p-3))
b <- cbind(b1,b2)
x <- matrix(rnorm(n*p), nrow = n, ncol = p)
y <- x %*% b1 + (x %*% b1)^3 + 4*(x %*% b2)^2 + rnorm(n)
MSIR <- msir(x, y)
summary(MSIR)
plot(MSIR)
plot(MSIR, which = 1:2)
plot(MSIR, type = "2Dplot", which = 1, span = 0.7)
plot(MSIR, type = "2Dplot", which = 2, span = 0.7)
plot(MSIR, type = "spinplot")
plot(MSIR, type = "evalues")
plot(MSIR, type = "coefficients")

## End(Not run)

Model-based Sliced Inverse Regression directions

Description

MSIR estimates a set of d \le p orthogonal direction vectors of length p which are estimates of the basis of the dimensional reduction subspace.

Usage

  ## S3 method for class 'msir'
predict(object, dim = 1:object$numdir, newdata, ...)

Arguments

object

an object of class 'msir' resulting from a call to msir.

dim

the dimensions of the reduced subspace used for prediction.

newdata

a data frame or matrix giving the data. If missing the data obtained from the call to msir are used.

...

further arguments passed to or from other methods.

Value

The function returns a matrix of points projected on the subspace spanned by the estimated basis vectors.

Author(s)

Luca Scrucca luca.scrucca@unipg.it

References

Scrucca, L. (2011) Model-based SIR for dimension reduction. Computational Statistics & Data Analysis, 55(11), 3010-3026.

See Also

{msir}

Examples

n <- 200
p <- 5
b <- as.matrix(c(1,-1,rep(0,p-2)))
x <- matrix(rnorm(n*p), nrow = n, ncol = p)
y <- exp(0.5 * x%*%b) + 0.1*rnorm(n)
pairs(cbind(y,x), gap = 0)

MSIR <- msir(x, y)
summary(MSIR)
plot(MSIR, which = 1, type = "2Dplot")
all.equal(predict(MSIR), MSIR$dir)
predict(MSIR, dim = 1:2)

x0 <- matrix(rnorm(n*p), nrow = n, ncol = p)
y0 <- exp(0.5 * x0%*%b) + 0.1*rnorm(n)
plot(predict(MSIR, dim = 1, newdata = x0), y0)

Rotating three-dimensional plot

Description

General function to draw a rgl-based rotating 3D scatterplot.

Usage

spinplot(x, y, z, 
         scaling = c("abc", "aaa"), 
         rem.lin.trend = FALSE, 
         uncor.vars = FALSE, 
         fit.ols = FALSE, 
         fit.smooth = FALSE, 
         span = 0.75, 
         ngrid = 25, 
         markby, 
         pch.points = 1, 
         col.points = "black", 
         cex.points = 1,
         col.axis   = "gray50", 
         col.smooth = "limegreen", 
         col.ols    = "lightsteelblue",
         background = "white", 
         ...)

Arguments

x

a vector of values for the variable in the horizontal (H) screen axis.

y

a vector of values for the variable in the vertical (V) screen axis.

z

a vector of values for the variable in the out-of-screen (O) axis.

scaling

the scaling applied. Two possible values are "abc" and "aaa".

rem.lin.trend

a logical specifying if the linear trend should be remove. If TRUE then the vertical axis is replaced by e(V|H,O), i.e. the residuals from a linear fit of the vertical axis variable on the others.

uncor.vars

a logical specifying if uncorrelated H and O variables should be used. If TRUE then the O variable is replaced by e(O|H), i.e. the residuals of the regression of O on H, hence obtaining a pair of uncorrelated variables.

fit.ols

a logical specifying if a fitted OLS plane should be included.

fit.smooth

a logical specifying if a nonparametric smoothing plane should be included.

span

the span used by loess to fit the polynomial surface.

ngrid

the number of grid points to use for displaing the fitted plane.

markby

a variable (usually a factor) to be used for marking the points.

pch.points

a vector of symbols for marking the points.

col.points

a vector of colors for marking the points.

cex.points

the cex for points.

col.axis

the color of the axis.

col.ols

the color to be used for drawing the OLS plane.

col.smooth

the color to be used for drawing the smoothing plane.

background

the color of background space.

...

catch further unused arguments.

Details

This function is mainly based on the functionality of the spin-plot function once available in XLisp-Stat software https://en.wikipedia.org/wiki/XLispStat, and the adds-on introduced by the Arc software http://www.stat.umn.edu/arc/index.html.

Author(s)

Luca Scrucca luca.scrucca@unipg.it

References

Cook R. D., Weisberg S. (1999) Applied Regression Including Computing and Graphics, Wiley, Chapter 8

Examples

## Not run: 
x1 <- rnorm(100)
x2 <- rnorm(100)
y  <- 2*x1 + x2^2 + 0.5*rnorm(100)
spinplot(x1, y, x2)
spinplot(x1, y, x2, scaling = "aaa")
spinplot(x1, y, x2, rem.lin.trend = "TRUE")
spinplot(x1, y, x2, fit.smooth = TRUE)
spinplot(x1, y, x2, fit.ols = TRUE)

x <- iris[,1:3]
y <- iris[,5]
spinplot(x)
spinplot(x, markby = y)
spinplot(x, markby = y, col.points = c("dodgerblue2", "orange", "green3"))
spinplot(x, markby = y, pch = c(0,3,1), col.points = c("dodgerblue2", "orange", "green3"))

# to save plots use 
# rgl.postscript("plot.pdf", fmt="pdf")
# or
# rgl.snapshot("plot.png")

## End(Not run)

Summary and print methods for 'msir' objects

Description

Summary and print methods for 'msir' objects.

Usage

## S3 method for class 'msir'
summary(object, numdir = object$numdir, std = FALSE, verbose = TRUE, ...)
## S3 method for class 'summary.msir'
print(x, digits = max(5, getOption("digits") - 3), ... )

Arguments

object

a 'msir' object

numdir

the number of directions to be shown.

std

if TRUE the coefficients basis are scaled such that all predictors have unit standard deviation.

verbose

if FALSE the coefficients basis are omitted; by default verbose = TRUE.

x

a 'summary.msir' object.

digits

the significant digits to use.

...

additional arguments.

Author(s)

Luca Scrucca luca.scrucca@unipg.it

See Also

msir