spatmca {SpatMCA}

Regularized spatial MCA

Description

Produce spatial coupled patterns at the designated locations according to the specified tuning parameters or the tuning parameters selected by M-fold cross-validation.

Usage

spatmca(
  x1,
  x2,
  Y1,
  Y2,
  M = 5,
  K = NULL,
  is_K_selected = ifelse(is.null(K), TRUE, FALSE),
  tau1u = NULL,
  tau2u = NULL,
  tau1v = NULL,
  tau2v = NULL,
  x1New = NULL,
  x2New = NULL,
  center = TRUE,
  maxit = 100,
  thr = 1e-04,
  are_all_tuning_parameters_selected = FALSE,
  num_cores = NULL
)

Arguments

x1

Location matrix (p \times d) corresponding to Y1. Each row is a location. d=1,2 is the dimension of locations.

x2

Location matrix (q \times d) corresponding to Y2. Each row is a location.

Y1

Data matrix (n \times p) of the first variable stores the values at p locations with sample size n.

Y2

Data matrix (n \times q) of the second variable stores the values at q locations with sample size n.

M

Optional number of folds; default is 5.

K

Optional user-supplied number of coupled patterns; default is NULL. If K is NULL or is_K_selected is TRUE, K is selected automatically.

is_K_selected

If TRUE, K is selected automatically; otherwise, is_K_selected is set to be user-supplied K. Default depends on user-supplied K.

tau1u

Optional user-supplied numeric vector of a nonnegative smoothness parameter sequence corresponding to Y1. If NULL, 10 tau1u values in a range are used.

tau2u

Optional user-supplied numeric vector of a nonnegative smoothness parameter sequence corresponding to Y1. If NULL, 10 tau2u values in a range are used.

tau1v

Optional user-supplied numeric vector of a nonnegative smoothness parameter sequence corresponding to Y2. If NULL, 10 tau1v values in a range are used.

tau2v

Optional user-supplied numeric vector of a nonnegative smoothness parameter sequence corresponding to Y2. If NULL, 10 tau2v values in a range are used.

x1New

New location matrix corresponding to Y1. If NULL, it is x1.

x2New

New location matrix corresponding to Y2. If NULL, it is x2.

center

If TRUE, center the columns of Y. Default is FALSE.

maxit

Maximum number of iterations. Default value is 100.

thr

Threshold for convergence. Default value is 10^{-4}.

are_all_tuning_parameters_selected

If TRUE, the K-fold CV performs to select 4 tuning parameters simultaneously. Default value is FALSE.

num_cores

Number of cores used to parallel computing. Default value is NULL (See RcppParallel::defaultNumThreads())

Details

The optimization problem is

\max_{\mathbf{U}, \mathbf{V}} \frac{1}{n}\mbox{tr}(\mathbf{U}'\mathbf{Y}'_1\mathbf{Y}_2\mathbf{V}) - \tau_{1u}\mbox{tr}(\mathbf{U}'\mathbf{\Omega}_1\mathbf{U}) - \tau_{2u}\sum_{k=1}^K\sum_{j=1}^{p} |u_{jk}|- \tau_{1v}\mbox{tr}(\mathbf{V}'\mathbf{\Omega}_2\mathbf{V})-\tau_{2v}\sum_{k=1}^K\sum_{j=1}^{q} |v_{jk}|,

\mbox{subject to $ \mathbf{U}'\mathbf{U}=\mathbf{V}'\mathbf{V}=\mathbf{I}_K$,} where \mathbf{Y}_1 and \mathbf{Y}_2 are two data matrices, {\mathbf{\Omega}}_1 and {\mathbf{\Omega}}_2 are two smoothness matrix, \mathbf{V}=\{v_{jk}\}, and \mathbf{U}=\{u_{jk}\}.

Value

A list of objects including

Uestfn

Estimated patterns for Y1 at the new locations, x1New.

Vestfn

Estimated patterns for Y2 at the new locations, x2New.

Dest

Estimated singular values.

crosscov

Estimated cross-covariance matrix between Y1 and Y2.

stau1u

Selected tau1u.

stau2u

Selected tau2u.

stau1v

Selected tau1v.

stau2v

Selected tau2v.

cv1

cv scores for tau1u and tau1v when are_all_tuning_parameters_selected is FALSE.

cv2

cv scores for tau2u and tau2v when are_all_tuning_parameters_selected is FALSE.

cvall

cv scores for tau1u, tau2u, tau1v and tau2v when are_all_tuning_parameters_selected is TRUE.

tau1u

Sequence of tau1u-values used in the process.

tau2u

Sequence of tau2u-values used in the process.

tau1v

Sequence of tau1v-values used in the process.

tau2v

Sequence of tau2v-values used in the process.

Author(s)

Wen-Ting Wang and Hsin-Cheng Huang

References

Wang, W.-T. and Huang, H.-C. (2017). Regularized principal component analysis for spatial data. Journal of Computational and Graphical Statistics 26 14-25.

Examples

originalPar <- par(no.readonly = TRUE)
# The following examples only use two threads for parallel computing.
## 1D: regular locations
p <- q <- 10
n <- 100
x1 <- matrix(seq(-7, 7, length = p), nrow = p, ncol = 1)
x2 <- matrix(seq(-7, 7, length = q), nrow = q, ncol = 1)
u <- exp(-x1^2) / norm(exp(-x1^2), "F")
v <- exp(-(x2 - 2)^2) / norm(exp(-(x2 - 2)^2), "F")
Sigma <- array(0, c(p + q, p + q))
Sigma[1:p, 1:p] <- diag(p)
Sigma[(p + 1):(p + q), (p + 1):(p + q)] <- diag(p)
Sigma[1:p, (p + 1):(p + q)] <- u %*% t(v)
Sigma[(p + 1):(p + q), 1:p] <- t(Sigma[1:p, (p + 1):(p + q)])
noise <- MASS::mvrnorm(n, mu = rep(0, p + q), Sigma = 0.001 * diag(p + q))
Y <- MASS::mvrnorm(n, mu = rep(0, p + q), Sigma = Sigma) + noise
Y1 <- Y[, 1:p]
Y2 <- Y[, -(1:p)]
cv1 <- spatmca(x1, x2, Y1, Y2, num_cores = 2)

par(mfrow = c(2, 1))
plot(x1, cv1$Uestfn[, 1], type='l', main = "1st pattern for Y1")
plot(x1, cv1$Vestfn[, 1], type='l', main = "1st pattern for Y2")
## Avoid changing the global enviroment
par(originalPar)


# The following examples will be executed more than 5 secs or including other libraries.
## 1D: artificial irregular locations
rmLoc1 <- sample(1:p, 3)
rmLoc2 <- sample(1:q, 4)
x1Rm <- x1[-rmLoc1]
x2Rm <- x2[-rmLoc2]
Y1Rm <- Y1[, -rmLoc1]
Y2Rm <- Y2[, -rmLoc2]
x1New <- as.matrix(seq(-7, 7, length = 100))
x2New <- as.matrix(seq(-7, 7, length = 50))
cv2 <- spatmca(x1 = x1Rm,
               x2 = x2Rm,
               Y1 = Y1Rm,
               Y2 = Y2Rm,
               x1New = x1New,
               x2New = x2New)
par(mfrow = c(2, 1))
plot(x1New, cv2$Uestfn[,1], type='l', main = "1st pattern for Y1")
plot(x2New, cv2$Vestfn[,1], type='l', main = "1st pattern for Y2")
par(originalPar)

## 2D real data
##  Daily 8-hour ozone averages and maximum temperature obtained from 28 monitoring
##  sites of NewYork, USA. It is of interest to see the relationship between the ozone
##  and the temperature through the coupled patterns.

library(spTimer)
library(pracma)
library(fields)
library(maps)
data(NYdata)
NYsite <- unique(cbind(NYdata[, 1:3]))
date <- as.POSIXct(seq(as.Date("2006-07-01"), as.Date("2006-08-31"), by = 1))
cMAXTMP<- matrix(NYdata[,8], 62, 28)
oz <- matrix(NYdata[,7], 62, 28)
rmNa <- !colSums(is.na(oz))
temp <- detrend(matrix(cMAXTMP[, rmNa], nrow = nrow(cMAXTMP)), "linear")
ozone <- detrend(matrix(oz[, rmNa], nrow = nrow(oz)), "linear")
x1 <- NYsite[rmNa, 2:3]
cv <- spatmca(x1, x1, temp, ozone)
par(mfrow = c(2, 1))
quilt.plot(x1, cv$Uestfn[, 1],
           xlab = "longitude",
           ylab = "latitude",
           main = "1st spatial pattern for temperature")
map(database = "state", regions = "new york", add = TRUE)
quilt.plot(x1, cv$Vestfn[, 1],
           xlab = "longitude",
           ylab = "latitude",
           main = "1st spatial pattern for ozone")
map(database = "state", regions = "new york", add = TRUE)
par(originalPar)

### Time series for the coupled patterns
tstemp <- temp %*% cv$Uestfn[,1]
tsozone <- ozone %*% cv$Vestfn[,1]
corr <- cor(tstemp, tsozone)
plot(date, tstemp / sd(tstemp), type='l', main = "Time series", ylab = "", xlab = "month")
lines(date, tsozone/sd(tsozone),col=2)
legend("bottomleft", c("Temperature (standardized)", "Ozone (standardized)"), col = 1:2, lty = 1:1)
mtext(paste("Pearson's correlation = ", round(corr, 3)), 3)

newP <- 50
xLon <- seq(-80, -72, length = newP)
xLat <- seq(41, 45, length = newP)
xxNew <- as.matrix(expand.grid(x = xLon, y = xLat))
cvNew <- spatmca(x1 = x1,
                 x2 = x1,
                 Y1 = temp,
                 Y2 = ozone,
                 K = cv$Khat,
                 tau1u = cv$stau1u,
                 tau1v = cv$stau1v,
                 tau2u = cv$stau2u,
                 tau2v = cv$stau2v,
                 x1New = xxNew,
                 x2New = xxNew)
par(mfrow = c(2, 1))
quilt.plot(xxNew, cvNew$Uestfn[, 1],
           nx = newP,
           ny = newP,
           xlab = "longitude",
           ylab = "latitude",
           main = "1st spatial pattern for temperature")
map(database = "county", regions = "new york", add = TRUE)
map.text("state", regions = "new york", cex = 2, add = TRUE)
quilt.plot(xxNew, cvNew$Vestfn[, 1],
           nx = newP,
           ny = newP,
           xlab = "longitude",
           ylab = "latitude",
           main = "2nd spatial pattern for ozone")
map(database = "county", regions = "new york", add = TRUE)
map.text("state", regions = "new york", cex = 2, add = TRUE)
par(originalPar)

## 3D: regular locations
n <- 200
x <- y <- z <- as.matrix(seq(-7, 7, length = 8))
d <- expand.grid(x, y, z)
u3D <- v3D <- exp(-d[, 1]^2 - d[, 2]^2 -d[, 3]^2)
p <- q <- 8^3
Sigma3D <- array(0, c(p + q, p + q))
Sigma3D[1:p, 1:p] <- diag(p)
Sigma3D[(p + 1):(p + q), (p + 1):(p + q)] <- diag(p)
Sigma3D[1:p, (p + 1):(p + q)] <- u3D %*% t(v3D)
Sigma3D[(p + 1):(p + q), 1:p] <- t(Sigma3D[1:p, (p + 1):(p + q)])

noise3D <- MASS::mvrnorm(n, mu = rep(0, p + q), Sigma = 0.001 * diag(p + q))
Y3D <- MASS::mvrnorm(n, mu = rep(0, p + q), Sigma = Sigma3D) + noise3D
Y13D <- Y3D[, 1:p]
Y23D <- Y3D[, -(1:p)]
cv3D <- spatmca(d, d, Y13D, Y23D)

library(plot3D)
library(RColorBrewer)
cols <- colorRampPalette(brewer.pal(9, 'Blues'))(10)
isosurf3D(x, y, z,
          colvar = array(cv3D$Uestfn[, 1], c(8, 8, 8)),
          level = seq(min(cv3D$Uestfn[, 1]), max(cv3D$Uestfn[, 1]), length = 10),
          ticktype = "detailed",
          colkey = list(side = 1),
          col = cols,
          main = "1st estimated pattern for Y1")

isosurf3D(x, y, z,
          colvar = array(cv3D$Vestfn[, 1], c(8, 8, 8)),
          level = seq(min(cv3D$Vestfn[, 1]), max(cv3D$Vestfn[,1]), length = 10),
          ticktype = "detailed",
          colkey = list(side = 1),
          col = cols,
          main = "1st estimated pattern for Y2")