Chpater 2: Functions for Fourier and Convolution Transformation Methods in Sufficient Dimension Reduction (SDR) Subspace Estimation
This section provides an overview of the functions within the
itdr package that utilize the Fourier transformation
method to estimate sufficient dimension reduction (SDR) subspaces in
regression. Specifically, we focus on the Fourier transformation method.
However, by passing the argument method="CM" to the
itdr() function, the convolution transformation method can
be employed.
Before estimating the SDR subspaces, it is essential to determine the
dimension (d) of the SDR subspace and tuning parameters
sw2, and st2. Section 2.1 demonstrates the
estimation of dimension (d) is demonstrated. The estimation of the
tuning parameter sw2 for both the central subspace (CS) and
the central mean subspace (CMS), is explained in Section 2.2.1.
Moreover, Section 2.2.2 outlines the estimation of st2 for
the central subspace (CS). Finally, The practical application of the
itdr() function for estimating the central subspace is provided in
Section 2.3.
2.1: Estimating the dimension (d) of sufficient dimension reduction (SDR) subspaces
A bootstrap estimation procedure is employed to estimate the unknown
dimension (d) of sufficient dimension reduction subspaces, as detailed
in Zhu and Zeng (2006). The d.boots() function can be used
to implement this estimation. To estimate the dimension (d)
of the central subspace of the automobile dataset, with
response variable \(y\), and predictor
variables \(x\) as specified in Zhu and
Zeng (2006), the following arguments are passed to the
d.boots() function: space="pdf" to estimate
the CS, xdensity="normal" for assuming normal density for
the predictors, and method="FM" for utilizing the Fourier
transformation method.
library(itdr)
data(automobile)
automobile.na <- na.omit(automobile)
# prepare response and predictor variables
auto_y <- log(automobile.na[, 26])
auto_xx <- automobile.na[, c(10, 11, 12, 13, 14, 17, 19, 20, 21, 22, 23, 24, 25)]
auto_x <- scale(auto_xx) # Standardize the predictors
# call to the d.boots() function with required #arguments
d_est <- d.boots(auto_y, auto_x,
var_plot = TRUE, space = "pdf",
xdensity = "normal", method = "FM"
)
#>
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Here, the estimate of the dimension of the central subspace for ‘automobile’ data is 2, i.e., d_hat=2.
2.2: Estimating tuning parameters and bandwidth parameters for Gaussian kernel density estimation
In the process of estimating SDR subspaces using the Fourier method,
two crucial parameters need to be determined: sw2 and
st2. The sw2. While sw2 is
required for both the central mean (CMS) and the central subspace (CS),
st2 is only necessary for the central subspace. The code in
Section 2.2.1 demonstrates the use of function wx() to
estimate the tuning parameter sw2, while Section 2.2.2
elaborates on using the wy() function to estimate the
tuning parameter st2.
2.2.1: Estimate sw2
To estimate the tuning parameter sw2, the
wx() function is utilized with the subspace option set to
either space="pdf" for the CS and space="mean"
for the CMS. Other parameters remain fixed during the estimation
process. The following R code chunk demonstrates the estimation of
sw2 for the central subspace.
auto_d <- 2 # The estimated value from Section 2.1
candidate_list<-seq(0.05, 1, by = 0.01)
auto_sw2 <- hyperPara(auto_y, auto_x, auto_d, range = candidate_list,xdensity = "normal", B = 500, space = "pdf", method = "FM", hyper = "wx")
#>
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auto_sw2$wx.hat # we get the estimator for sw2 as 0.09
#> [1] 0.052.2.2: Estimate st2
The estimate of the tuning parameter st2, the
wy() function is employed. Other parameters are fixed
during the estimation. Notice that, there is no need to specify the
space, because the tuning parameter st2
exclusively required for the central subspace (CS).
auto_d <- 2 # Estimated value from Section 2.1
auto_st2 <- hyperPara(auto_y, auto_x, auto_d, wx = 0.1, range = seq(0.1, 1, by = 0.1), xdensity = "normal", method = "FM", hyper = "wy")
#>
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auto_st2$wy.hat # we get the estimator for st2=0.9
#> [1] 0.82.2.3: Estimate the bandwidth (h) of the Gaussian
kernel density function
When the distribution function of the predictor variables is unknown,
Gaussian kernel density estimation is utilized to approximate the
density function of the predictor variables. In such cases, the
bandwidth parameter needs to be estimated, especially when
xdensity="kernel" is specified. The wh()
function uses the bootstrap estimator to estimate the bandwidth of the
Gaussian kernel density estimation.
h_hat <- hyperPara(auto_y, auto_x, auto_d, wx = 5, wy = 0.1, range = seq(0.1, 2, by = .1), space = "pdf", method = "FM", hyper = "wh")
#>
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# Bandwidth estimator for Gaussian kernel density estimation for central subspace
h_hat$h.hat # we have the estimator as h_hat=0.1
#> [1] 0.2Ensure proper parameterization and selection of options for accurate estimation.
2.3: Estimate SDR subspaces
Having outlined the estimation procedure for tuning parameters in the
Fourier method in Sections 2.1-2.2, we are now prepared to estimate the
SDR subspaces. Zhu and Zeng (2006) utilized the Fourier method to
facilitate the estimation of the SDR subspaces when the predictors
follow a multivariate normal distribution. However, when the predictor
variables follow an elliptical distribution or, more generally, when the
distribution of the predictors is unknown, the predictors’ distribution
function is approximated using Gaussian kernel density estimation (Zeng
and Zhu, 2010). The itdr() function is employed to estimate
the SDR subspaces under the FM method as follows. Since the
default setting of the itdr() function has
method="FM", it is optional to specify the method as
“FM”.
library(itdr)
data(automobile)
head(automobile)
#> symboling normalized make fuelType aspiration #doors bodyStyle
#> 1 3 NA alfa-romero gas std two convertible
#> 2 3 NA alfa-romero gas std two convertible
#> 3 1 NA alfa-romero gas std two hatchback
#> 4 2 164 audi gas std four sedan
#> 5 2 164 audi gas std four sedan
#> 6 2 NA audi gas std two sedan
#> driveWheel engineLoc wheelBase length width height curbWeight engineType
#> 1 rwd front 88.6 168.8 64.1 48.8 2548 dohc
#> 2 rwd front 88.6 168.8 64.1 48.8 2548 dohc
#> 3 rwd front 94.5 171.2 65.5 52.4 2823 ohcv
#> 4 fwd front 99.8 176.6 66.2 54.3 2337 ohc
#> 5 4wd front 99.4 176.6 66.4 54.3 2824 ohc
#> 6 fwd front 99.8 177.3 66.3 53.1 2507 ohc
#> #Cylinder engineSize fuelSys bore stroke compession norsepower peak_rpm
#> 1 four 130 mpfi 3.47 2.68 9.0 111 5000
#> 2 four 130 mpfi 3.47 2.68 9.0 111 5000
#> 3 six 152 mpfi 2.68 3.47 9.0 154 5000
#> 4 four 109 mpfi 3.19 3.40 10.0 102 5500
#> 5 five 136 mpfi 3.19 3.40 8.0 115 5500
#> 6 five 136 mpfi 3.19 3.40 8.5 110 5500
#> city_rpm highwayMPG price
#> 1 21 27 13495
#> 2 21 27 16500
#> 3 19 26 16500
#> 4 24 30 13950
#> 5 18 22 17450
#> 6 19 25 15250
df <- cbind(automobile[, c(26, 10, 11, 12, 13, 14, 17, 19, 20, 21, 22, 23, 24, 25)])
dff <- as.matrix(df)
automobi <- dff[complete.cases(dff), ]
d <- 2
# Estimated value from Section 2.1
wx <- .14 # Estimated value from Section 2.2.1
wy <- .9 # Estimated value from Section 2.2.2
wh <- 1.5 # Estimated value from Section 2.2.3
p <- 13 # Estimated value from Section 2.3
y <- automobi[, 1]
x <- automobi[, c(2:14)]
xt <- scale(x)
# Distribution of the predictors is a normal distribution
fit.F_CMS <- itdr(y, xt, d, wx, wy, wh, space = "pdf", xdensity = "normal", method = "FM")
round(fit.F_CMS$eta_hat, 2)
#> [,1] [,2]
#> [1,] -0.10 0.33
#> [2,] 0.26 -0.19
#> [3,] 0.07 -0.02
#> [4,] -0.01 -0.13
#> [5,] -0.57 0.28
#> [6,] 0.22 -0.37
#> [7,] -0.04 -0.11
#> [8,] 0.16 0.23
#> [9,] -0.20 -0.15
#> [10,] -0.31 0.18
#> [11,] -0.08 -0.26
#> [12,] -0.49 0.50
#> [13,] 0.36 -0.43
# Distribution of the predictors is a unknown (using kernel method)
fit.F_CMS <- itdr(y, xt, d, wx, wy, wh, space = "pdf", xdensity = "kernel", method = "FM")
round(fit.F_CMS$eta_hat, 2)
#> [,1] [,2]
#> [1,] 0.09 -0.04
#> [2,] 0.01 0.07
#> [3,] -0.01 0.02
#> [4,] 0.01 0.00
#> [5,] -0.19 -0.10
#> [6,] 0.26 0.89
#> [7,] 0.06 -0.32
#> [8,] -0.14 -0.18
#> [9,] -0.89 0.03
#> [10,] -0.07 -0.02
#> [11,] 0.15 0.04
#> [12,] 0.19 0.20
#> [13,] -0.12 -0.10
