Calculate the potential of the points w.r.t. a multivariate data set. The potential is the kernel-estimated density multiplied by the prior probability of a class. Different from the data depths, a density estimate measures at a given point how much mass is located around it.
depth.potential (x, data, pretransform = "1Mom",
kernel = "GKernel", kernel.bandwidth = NULL, mah.parMcd = 0.75)
Numerical vector of potentials, one for each row in x
; or one potential value if x
is a numerical vector.
Matrix of objects (numerical vector as one object) whose depth is to be calculated; each row contains a data
.
Matrix of data where each row contains a
The method of data scaling.
NULL
to use the original data,
1Mom
or NMom
for scaling using data moments,
1MCD
or NMCD
for scaling using robust data moments (Minimum Covariance Determinant (MCD) ).
"EDKernel"
for the kernel of type 1/(1+kernel.bandwidth*EuclidianDistance2(x, y)),
"GKernel"
[default and recommended] for the simple Gaussian kernel,
"EKernel"
exponential kernel: exp(-kernel.bandwidth*EuclidianDistance(x, y)),
"VarGKernel"
variable Gaussian kernel, where kernel.bandwidth
is proportional to the depth.zonoid
of a point.
the single bandwidth parameter of the kernel. If NULL
- the Scott's rule of thumb is used.
is the value of the argument alpha
for the function covMcd
; is used when pretransform = "*MCD"
.
The potential is the kernel-estimated density multiplied by the prior probability of a class. The kernel bandwidth matrix is decomposed into two parts, one of which describes the form of the data, and the other the width of the kernel. Then the first part is used to transform the data using the moments, while the second is employed as a parameter of the kernel and tuned to achieve the best separation. For details see Pokotylo and Mosler (2015).
Aizerman, M.A., Braverman, E.M., and Rozonoer, L.I. (1970). The Method of Potential Functions in the Theory of Machine Learning. Nauka (Moscow).
Pokotylo, O. and Mosler, K. (2015). Classification with the pot-pot plot. Mimeo.
depth.halfspace
for calculation of the Tukey depth.
depth.Mahalanobis
for calculation of Mahalanobis depth.
depth.projection
for calculation of projection depth.
depth.simplicial
for calculation of simplicial depth.
depth.simplicialVolume
for calculation of simplicial volume depth.
depth.spatial
for calculation of spatial depth.
depth.zonoid
for calculation of zonoid depth.
# 3-dimensional normal distribution
data <- mvrnorm(200, rep(0, 3),
matrix(c(1, 0, 0,
0, 2, 0,
0, 0, 1),
nrow = 3))
x <- mvrnorm(10, rep(1, 3),
matrix(c(1, 0, 0,
0, 1, 0,
0, 0, 1),
nrow = 3))
# potential with rule of thumb bandwidth
pot <- depth.potential(x, data)
cat("Potentials: ", pot, "\n")
# potential with bandwidth = 0.1
pot <- depth.potential(x, data, kernel.bandwidth = 0.1)
cat("Potentials: ", pot, "\n")
# potential with robust MCD scaling
pot <- depth.potential(x, data, kernel.bandwidth = 0.1,
pretransform = "NMCD", mah.parMcd = 0.6)
cat("Potentials: ", pot, "\n")
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