sid {resemble} | R Documentation |
A function for computing the spectral information divergence between spectra (sid)
Description
This function computes the spectral information divergence/dissimilarity between spectra based on the kullback-leibler divergence algorithm (see details).
Usage
sid(Xr, Xu = NULL,
mode = "density",
center = FALSE, scale = FALSE,
kernel = "gaussian",
n = if(mode == "density") round(0.5 * ncol(Xr)),
bw = "nrd0",
reg = 1e-04,
...)
Arguments
Xr |
a matrix containing the spectral (reference) data. |
Xu |
an optional matrix containing the spectral data of a second set of observations. |
mode |
the method to be used for computing the spectral information
divergence. Options are |
center |
a logical indicating if the computations must be carried out on
the centred |
scale |
a logical indicating if the computations must be carried out on
the variance scaled |
kernel |
if |
n |
if |
bw |
if |
reg |
a numerical value larger than 0 which indicates a regularization parameter. Values (probabilities) below this threshold are replaced by this value for numerical stability. Default is 1e-4. |
... |
additional arguments to be passed to the
|
Details
This function computes the spectral information divergence (distance)
between spectra.
When mode = "density"
, the function first computes the probability
distribution of each spectrum which result in a matrix of density
distribution estimates. The density distributions of all the observations in
the datasets are compared based on the kullback-leibler divergence algorithm.
When mode = "feature"
, the kullback-leibler divergence between all
the observations is computed directly on the spectral variables.
The spectral information divergence (SID) algorithm (Chang, 2000) uses the
Kullback-Leibler divergence (\(KL\)) or relative entropy
(Kullback and Leibler, 1951) to account for the vis-NIR information provided
by each spectrum. The SID between two spectra (\(x_{i}\) and
\(x_{j}\)) is computed as follows:
where \(k\) represents the number of variables or spectral features, \(p\) and \(q\) are the probability vectors of \(x_{i}\) and \(x_{i}\) respectively which are calculated as:
\[p = \frac{x_i}{\sum_{l=1}^{k} x_{i,l}}\] \[q = \frac{x_j}{\sum_{l=1}^{k} x_{j,l}}\]From the above equations it can be seen that the original SID algorithm
assumes that all the components in the data matrices are nonnegative.
Therefore centering cannot be applied when mode = "feature"
. If a
data matrix with negative values is provided and mode = "feature"
,
the sid
function automatically scales the matrix as follows:
or
\[X_{s} = \frac{X-min(X, Xu)}{max(X, Xu)-min(X, Xu)}\] \[Xu_{s} = \frac{Xu-min(X, Xu)}{max(X, Xu)-min(X, Xu)}\]if Xu
is specified. The 0 values are replaced by a regularization
parameter (reg
argument) for numerical stability.
The default of the sid
function is to compute the SID based on the
density distributions of the spectra (mode = "density"
). For each
spectrum in X
the density distribution is computed using the
density
function of the stats
package.
The 0 values of the estimated density distributions of the spectra are
replaced by a regularization parameter ("reg"
argument) for numerical
stability. Finally the divergence between the computed spectral histogramas
is computed using the SID algorithm. Note that if mode = "density"
,
the sid
function will accept negative values and matrix centering
will be possible.
Value
a list
with the following components:
sid
: if only"X"
is specified (i.e.Xu = NULL
), a square symmetric matrix of SID distances between all the components in"X"
. If both"X"
and"Xu"
are specified, a matrix of SID distances between the components in"X"
and the components in"Xu"
) where the rows represent the objects in"X"
and the columns represent the objects in"Xu"
Xr
: the (centered and/or scaled if specified) spectralX
matrixXu
: the (centered and/or scaled if specified) spectralXu
matrixdensityDisXr
: ifmode = "density"
, the computed density distributions ofXr
densityDisXu
: ifmode = "density"
, the computed density distributions ofXu
Author(s)
Leonardo Ramirez-Lopez
References
Chang, C.I. 2000. An information theoretic-based approach to spectral variability, similarity and discriminability for hyperspectral image analysis. IEEE Transactions on Information Theory 46, 1927-1932.
See Also
Examples
library(prospectr)
data(NIRsoil)
Xu <- NIRsoil$spc[!as.logical(NIRsoil$train), ]
Yu <- NIRsoil$CEC[!as.logical(NIRsoil$train)]
Yr <- NIRsoil$CEC[as.logical(NIRsoil$train)]
Xr <- NIRsoil$spc[as.logical(NIRsoil$train), ]
Xu <- Xu[!is.na(Yu), ]
Xr <- Xr[!is.na(Yr), ]
# Example 1
# Compute the SID distance between all the observations in Xr
xr_sid <- sid(Xr)
xr_sid
# Example 2
# Compute the SID distance between the observations in Xr and the observations
# in Xu
xr_xu_sid <- sid(Xr, Xu)
xr_xu_sid