Model-based Sliced Inverse Regression (MSIR)

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.

References

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

Author

Luca Scrucca luca.scrucca@unipg.it

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)
#> -------------------------------------------------- 
#> Model-based SIR 
#> -------------------------------------------------- 
#> 
#> Slices:
#>           1   2    3   4       5   6   
#> GMM       XII EEE  XXX EEI     XXX VII 
#> Num.comp. 1   2    1   3       1   2   
#> Num.obs.  33  26|7 33  15|6|12 33  28|7
#> 
#> Estimated basis vectors:
#>          Dir1     Dir2     Dir3     Dir4       Dir5
#> x1  0.7097864 -0.48270  0.53769  0.12034  0.0022288
#> x2 -0.7041481 -0.42937  0.45878  0.13769 -0.1827196
#> x3  0.0111170  0.71477  0.68775 -0.22556 -0.0522356
#> x4  0.0052141  0.16238  0.16504  0.60551  0.7581749
#> x5 -0.0150997 -0.21300 -0.01333 -0.74097  0.6237394
#> 
#>                 Dir1     Dir2     Dir3      Dir4       Dir5
#> Eigenvalues  0.91016  0.26585  0.15332  0.060237   0.012936
#> Cum. %      64.89545 83.85069 94.78262 99.077612 100.000000
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)
#> -------------------------------------------------- 
#> Model-based SIR 
#> -------------------------------------------------- 
#> 
#> Slices:
#>           1     2   3   4   5     6    
#> GMM       VVE   XII XII XII EEE   EEE  
#> Num.comp. 2     1   1   1   2     2    
#> Num.obs.  20|13 33  33  33  14|19 22|13
#> 
#> Estimated basis vectors:
#>         Dir1     Dir2      Dir3     Dir4     Dir5
#> x1 -0.740396 -0.39489 -0.247911 -0.45435 -0.37276
#> x2  0.666642 -0.41304 -0.083044 -0.49875 -0.29250
#> x3  0.068615 -0.65422  0.176128  0.67171 -0.33899
#> x4 -0.022374 -0.42329  0.596660 -0.23544  0.62061
#> x5  0.046850 -0.25746 -0.737983  0.19544  0.52481
#> 
#>                 Dir1      Dir2      Dir3       Dir4      Dir5
#> Eigenvalues  0.76871  0.053058  0.031793  0.0088756 6.346e-03
#> Cum. %      88.48121 94.588434 98.247933 99.2695486 1.000e+02
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)
#> -------------------------------------------------- 
#> Model-based SIR 
#> -------------------------------------------------- 
#> 
#> Slices:
#>           1     2     3    4   5     6     7     8  
#> GMM       EEI   EVE   VEI  XII EEV   VEV   EEV   XXI
#> Num.comp. 2     2     2    1   2     2     2     1  
#> Num.obs.  20|22 26|16 9|33 42  18|24 18|24 28|14 6  
#> 
#> Estimated basis vectors:
#>          Dir1      Dir2      Dir3     Dir4      Dir5
#> x1 -0.3884439  0.939737 -0.101742 0.147305  0.026227
#> x2  0.6952440  0.215864 -0.613699 0.048259  0.063465
#> x3  0.6036119  0.209895  0.776560 0.059747 -0.091979
#> x4  0.0369820 -0.161043 -0.093403 0.985688 -0.065283
#> x5  0.0056592  0.017512 -0.035387 0.028667 -0.991243
#> 
#>                 Dir1     Dir2     Dir3      Dir4       Dir5
#> Eigenvalues  0.71463  0.49623  0.29359  0.071781   0.047476
#> Cum. %      44.01218 74.57402 92.65524 97.076059 100.000000
plot(MSIR, which = 1:2)

if (FALSE) plot(MSIR, type = "spinplot")
plot(MSIR, which = 1, type = "2Dplot", span = 0.7)

plot(MSIR, which = 2, type = "2Dplot", span = 0.7)