select_h: Select the value of the kernel tuning parameter

In QuadratiK: Collection of Methods Constructed using Kernel-Based Quadratic Distances

Select the value of the kernel tuning parameter

Description

This function computes the kernel bandwidth of the Gaussian kernel for the normality, two-sample and k-sample kernel-based quadratic distance (KBQD) tests.

Usage

select_h(
  x,
  y = NULL,
  alternative = NULL,
  method = "subsampling",
  b = 0.8,
  B = 100,
  delta_dim = 1,
  delta = NULL,
  h_values = NULL,
  Nrep = 50,
  n_cores = 2,
  Quantile = 0.95,
  power.plot = TRUE
)

Arguments

x	Data set of observations from X.
y	Numeric matrix or vector of data values. Depending on the input y, the selection of h is performed for the corresponding test. if y = NULL, the function performs the tests for normality on x. if y is a data matrix, with same dimensions of x, the function performs the two-sample test between x and y. if y is a numeric or factor vector, indicating the group memberships for each observation, the function performs the k-sample test.
alternative	Family of alternative chosen for selecting h, between "location", "scale" and "skewness".
method	The method used for critical value estimation ("subsampling", "bootstrap", or "permutation").
b	The size of the subsamples used in the subsampling algorithm .
B	The number of iterations to use for critical value estimation, B = 150 as default.
delta_dim	Vector of coefficient of alternative with respect to each dimension
delta	Vector of parameter values indicating chosen alternatives
h_values	Values of the tuning parameter used for the selection
Nrep	Number of bootstrap/permutation/subsampling replications.
n_cores	Number of cores used to parallel the h selection algorithm. If this is not provided, the function will detect the available cores.
Quantile	The quantile to use for critical value estimation, 0.95 is the default value.
power.plot	Logical. If TRUE, it is displayed the plot of power for values in h_values and delta.

Details

The function performs the selection of the optimal value for the tuning parameter h of the normal kernel function, for normality test, the two-sample and k-sample KBQD tests. It performs a small simulation study, generating samples according to the family of alternative specified, for the chosen values of h_values and delta.

We consider target alternatives F_\delta(\hat{\mathbf{\mu}}, \hat{\mathbf{\Sigma}}, \hat{\mathbf{\lambda}}), where \hat{\mathbf{\mu}}, \hat{\mathbf{\Sigma}} and \hat{\mathbf{\lambda}} indicate the location, covariance and skewness parameter estimates from the pooled sample.

• Compute the estimates of the mean \hat{\mu}, covariance matrix \hat{\Sigma} and skewness \hat{\lambda} from the pooled sample.

• Choose the family of alternatives F_\delta = F_\delta(\hat{\mu} ,\hat{\Sigma}, \hat{\lambda}).
  
  For each value of \delta and h:

• Generate \mathbf{X}_1,\ldots,\mathbf{X}_{k-1} \sim F_0, for \delta=0;

• Generate \mathbf{X}_k \sim F_\delta;

• Compute the k-sample test statistic between \mathbf{X}_1, \mathbf{X}_2, \ldots, \mathbf{X}_k with kernel parameter h;

• Compute the power of the test. If it is greater than 0.5, select h as optimal value.

• If an optimal value has not been selected, choose the h which corresponds to maximum power.

The available alternative are
location alternatives, F_\delta = SN_d(\hat{\mu} + \delta,\hat{\Sigma}, \hat{\lambda}),with \delta = 0.2, 0.3, 0.4;
scale alternatives, F_\delta = SN_d(\hat{\mu} ,\hat{\Sigma}*\delta, \hat{\lambda}), \delta = 0.1, 0.3, 0.5;
skewness alternatives, F_\delta = SN_d(\hat{\mu} ,\hat{\Sigma}, \hat{\lambda} + \delta), with \delta = 0.2, 0.3, 0.6.
The values of h = 0.6, 1, 1.4, 1.8, 2.2 and N=50 are set as default values.
The function select_h() allows the user to set the values of \delta and h for a more extensive grid search. We suggest to set a more extensive grid search when computational resources permit.

Value

A list with the following attributes:

• h_sel the selected value of tuning parameter h;

• power matrix of power values computed for the considered values of delta and h_values;

• power.plot power plots (if power.plot is TRUE).

Note

Please be aware that the select_h() function may take a significant amount of time to run, especially with larger datasets or when using an larger number of parameters in h_values and delta. Consider this when applying the function to large or complex data.

References

Markatou, M. and Saraceno, G. (2024). “A Unified Framework for Multivariate Two- and k-Sample Kernel-based Quadratic Distance Goodness-of-Fit Tests.”
https://doi.org/10.48550/arXiv.2407.16374

Saraceno, G., Markatou, M., Mukhopadhyay, R. and Golzy, M. (2024). Goodness-of-Fit and Clustering of Spherical Data: the QuadratiK package in R and Python.
https://arxiv.org/abs/2402.02290.

See Also

The function select_h is used in the kb.test() function.

Examples

# Select the value of h using the mid-power algorithm

x <- matrix(rnorm(100), ncol = 2)
y <- matrix(rnorm(100), ncol = 2)
h_sel <- select_h(x, y, "skewness")
h_sel

QuadratiK documentation built on April 12, 2025, 2 a.m.