Create a Gaussian kernel matrix from covariates

nmfkc.kernel.gaussian constructs a Gaussian (RBF) kernel matrix from covariate matrices. The kernel is defined as \(K(u,v) = \exp(-\beta \|u - v\|^2)\). When V contains NA values, two methods are available via na.method:

"pds"

Partial Distance Strategy. Computes the kernel using only observed (non-NA) rows, with beta adjusted by \(\beta_{adj} = \beta \times K / K_{obs}\) where \(K\) is the total number of rows and \(K_{obs}\) is the number of observed rows.

"egk"

Expected Gaussian Kernel (Mesquita et al., 2019). Uses a Gaussian Mixture Model (GMM) to estimate the conditional distribution of missing values given observed values, then computes the expected kernel value via a Gamma approximation. Requires gmm.means, gmm.sigmas, and gmm.weights passed through ....

Usage

nmfkc.kernel.gaussian(
  U,
  V = NULL,
  beta = 0.5,
  na.method = c("pds", "egk"),
  ...
)

Source

Mesquita, D., Gomes, J. P., & Rodrigues, L. R. (2019). Gaussian kernels for incomplete data. Applied Soft Computing, 77, 356–365.

Arguments

U

Covariate matrix \(U(K,N) = (u_1, \dots, u_N)\). Each row may be normalized in advance.

V

Covariate matrix \(V(K,M) = (v_1, \dots, v_M)\), typically used for prediction. If NULL, the default is U. May contain NA values.

beta

Bandwidth parameter for the Gaussian kernel. Default is 0.5.

na.method

Method for handling NA values in V. Either "pds" or "egk". Ignored if V has no NA.

...

Additional arguments for EGK method:

gmm.G

Number of GMM components for EGK. Default is 3 (Mesquita et al., 2019).

Value

Kernel matrix \(A(N,M)\).

See also

nmfkc.kernel, nmfkc.kernel.beta.cv, nmfkc.kernel.beta.nearest.med

Examples

U <- matrix(c(5,10,15,20,25),nrow=1)
V <- matrix(1:25,nrow=1)
A <- nmfkc.kernel.gaussian(U,V,beta=28/1000)
dim(A)
#> [1]  5 25

# PDS example: V with NA in first row
U2 <- matrix(rnorm(20), nrow=2)
V2 <- matrix(rnorm(10), nrow=2)
V2[1, c(2,4)] <- NA
A2 <- nmfkc.kernel.gaussian(U2, V2, beta=0.5, na.method="pds")