Package 'DRR'

Dimensionality Reduction via Regression.

Description

DRR implements the Dimensionality Reduction via Regression using Kernel Ridge Regression. It also adds a faster implementation of Kernel Ridge regression that can be used with the CVST package.

Details

Funding provided by the Department for Biogeochemical Integration, Empirical Inference of the Earth System Group, at the Max Plack Institute for Biogeochemistry, Jena.

Author(s)

Maintainer: Guido Kraemer [email protected]

References

Laparra, V., Malo, J., Camps-Valls, G., 2015. Dimensionality Reduction via Regression in Hyperspectral Imagery. IEEE Journal of Selected Topics in Signal Processing 9, 1026-1036. doi:10.1109/JSTSP.2015.2417833 Zhang, Y., Duchi, J.C., Wainwright, M.J., 2013. Divide and Conquer Kernel Ridge Regression: A Distributed Algorithm with Minimax Optimal Rates. arXiv:1305.5029 [cs, math, stat].

See Also

Useful links:

• https://github.com/gdkrmr/DRR

• Report bugs at https://github.com/gdkrmr/DRR/issues

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Fast implementation for Kernel Ridge Regression.

Description

Constructs a learner for the divide and conquer version of KRR.

Usage

constructFastKRRLearner()

Details

This function is to be used with the CVST package as a drop in replacement for constructKRRLearner. The implementation approximates the inversion of the kernel Matrix using the divide an conquer scheme, lowering computational and memory complexity from $O(n^3)$ and $O(n^2)$ to $O(n^3/m^2)$ and $O(n^2/m^2)$ respectively, where m are the number of blocks to be used (parameter nblocks). Theoretically safe values for $m$ are $< n^{1/3}$, but practically $m$ may be a little bit larger. The function will issue a warning, if the value for $m$ is too large.

Value

Returns a learner similar to constructKRRLearner suitable for the use with CV and fastCV.

References

Zhang, Y., Duchi, J.C., Wainwright, M.J., 2013. Divide and Conquer Kernel Ridge Regression: A Distributed Algorithm with Minimax Optimal Rates. arXiv:1305.5029 [cs, math, stat].

See Also

constructLearner

Examples

ns <- noisySinc(1000)
nsTest <- noisySinc(1000)

fast.krr <- constructFastKRRLearner()
fast.p <- list(kernel="rbfdot", sigma=100, lambda=.1/getN(ns), nblocks = 4)
system.time(fast.m <- fast.krr$learn(ns, fast.p))
fast.pred <- fast.krr$predict(fast.m, nsTest)
sum((fast.pred - nsTest$y)^2) / getN(nsTest)

## Not run: 
krr <- CVST::constructKRRLearner()
p <- list(kernel="rbfdot", sigma=100, lambda=.1/getN(ns))
system.time(m <- krr$learn(ns, p))
pred <- krr$predict(m, nsTest)
sum((pred - nsTest$y)^2) / getN(nsTest)

plot(ns, col = '#00000030', pch = 19)
lines(sort(nsTest$x), fast.pred[order(nsTest$x)], col = '#00C000', lty = 2)
lines(sort(nsTest$x), pred[order(nsTest$x)], col = '#0000C0', lty = 2)
legend('topleft', legend = c('fast KRR', 'KRR'),
       col = c('#00C000', '#0000C0'), lty = 2)

## End(Not run)

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Dimensionality Reduction via Regression

Description

drr Implements Dimensionality Reduction via Regression using Kernel Ridge Regression.

Usage

drr(
  X,
  ndim = ncol(X),
  lambda = c(0, 10^(-3:2)),
  kernel = "rbfdot",
  kernel.pars = list(sigma = 10^(-3:4)),
  pca = TRUE,
  pca.center = TRUE,
  pca.scale = FALSE,
  fastcv = FALSE,
  cv.folds = 5,
  fastcv.test = NULL,
  fastkrr.nblocks = 4,
  verbose = TRUE
)

Arguments

X	input data, a matrix.
ndim	the number of output dimensions and regression functions to be estimated, see details for inversion.
lambda	the penalty term for the Kernel Ridge Regression.
kernel	a kernel function or string, see kernel-class for details.
kernel.pars	a list with parameters for the kernel. each parameter can be a vector, crossvalidation will choose the best combination.
pca	logical, do a preprocessing using pca.
pca.center	logical, center data before applying pca.
pca.scale	logical, scale data before applying pca.
fastcv	if TRUE uses fastCV, if FALSE uses CV for crossvalidation.
cv.folds	if using normal crossvalidation, the number of folds to be used.
fastcv.test	an optional separate test data set to be used for fastCV, handed over as option test to fastCV.
fastkrr.nblocks	the number of blocks used for fast KRR, higher numbers are faster to compute but may introduce numerical inaccurracies, see constructFastKRRLearner for details.
verbose	logical, should the crossvalidation report back.

Details

Parameter combination will be formed and cross-validation used to select the best combination. Cross-validation uses CV or fastCV.

Pre-treatment of the data using a PCA and scaling is made $\alpha = Vx$. the representation in reduced dimensions is

$y_i = \alpha - f_i(\alpha_1, \ldots, \alpha_{i-1})$

then the final DRR representation is:

$r = (\alpha_1, y_2, y_3, \ldots,y_d)$

DRR is invertible by

$\alpha_i = y_i + f_i(\alpha_1,\alpha_2, \ldots, alpha_{i-1})$

If less dimensions are estimated, there will be less inverse functions and calculating the inverse will be inaccurate.

Value

A list the following items:

• "fitted.data" The data in reduced dimensions.

• "pca.means" The means used to center the original data.

• "pca.scale" The standard deviations used to scale the original data.

• "pca.rotation" The rotation matrix of the PCA.

• "models" A list of models used to estimate each dimension.

• "apply" A function to fit new data to the estimated model.

• "inverse" A function to untransform data.

References

Laparra, V., Malo, J., Camps-Valls, G., 2015. Dimensionality Reduction via Regression in Hyperspectral Imagery. IEEE Journal of Selected Topics in Signal Processing 9, 1026-1036. doi:10.1109/JSTSP.2015.2417833

Examples

tt <- seq(0,4*pi, length.out = 200)
helix <- cbind(
  x = 3 * cos(tt) + rnorm(length(tt), sd = seq(0.1, 1.4, length.out = length(tt))),
  y = 3 * sin(tt) + rnorm(length(tt), sd = seq(0.1, 1.4, length.out = length(tt))),
  z = 2 * tt      + rnorm(length(tt), sd = seq(0.1, 1.4, length.out = length(tt)))
)
helix <- helix[sample(nrow(helix)),] # shuffling data is important!!
system.time(
drr.fit  <- drr(helix, ndim = 3, cv.folds = 4,
                lambda = 10^(-2:1),
                kernel.pars = list(sigma = 10^(0:3)),
                fastkrr.nblocks = 2, verbose = TRUE,
                fastcv = FALSE)
)

## Not run: 
library(rgl)
plot3d(helix)
points3d(drr.fit$inverse(drr.fit$fitted.data[,1,drop = FALSE]), col = 'blue')
points3d(drr.fit$inverse(drr.fit$fitted.data[,1:2]),             col = 'red')

plot3d(drr.fit$fitted.data)
pad <- -3
fd <- drr.fit$fitted.data
xx <- seq(min(fd[,1]),       max(fd[,1]),       length.out = 25)
yy <- seq(min(fd[,2]) - pad, max(fd[,2]) + pad, length.out = 5)
zz <- seq(min(fd[,3]) - pad, max(fd[,3]) + pad, length.out = 5)

dd <- as.matrix(expand.grid(xx, yy, zz))
plot3d(helix)
for(y in yy) for(x in xx)
  rgl.linestrips(drr.fit$inverse(cbind(x, y, zz)), col = 'blue')
for(y in yy) for(z in zz)
  rgl.linestrips(drr.fit$inverse(cbind(xx, y, z)), col = 'blue')
for(x in xx) for(z in zz)
  rgl.linestrips(drr.fit$inverse(cbind(x, yy, z)), col = 'blue')

## End(Not run)

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