NGPPsim {ICtest}

Signal Subspace Dimension Testing Using non-Gaussian Projection Pursuit

Description

Tests whether the true dimension of the signal subspace is less than or equal to a given k. The test statistic is a multivariate extension of the classical Jarque-Bera statistic and the distribution of it under the null hypothesis is obtained by simulation.

Usage

NGPPsim(X, k, nl = c("skew", "pow3"), alpha = 0.8, N = 1000, eps = 1e-6,
        verbose = FALSE, maxiter = 100)

Arguments

X

Numeric matrix with n rows corresponding to the observations and p columns corresponding to the variables.

k

Number of components to estimate, 1 \leq k \leq p.

nl

Vector of non-linearities, a convex combination of the corresponding squared objective functions of which is then used as the projection index. The choices include "skew" (skewness), "pow3" (excess kurtosis), "tanh" (log(cosh)) and "gauss" (Gaussian function).

alpha

Vector of positive weights between 0 and 1 given to the non-linearities. The length of alpha should be either one less than the number of non-linearities in which case the missing weight is chosen so that alpha sums to one, or equal to the number of non-linearities in which case the weights are used as such. No boundary checks for the weights are done.

N

Number of normal samples to be used in simulating the distribution of the test statistic under the null hypothesis.

eps

Convergence tolerance.

verbose

If TRUE the numbers of iterations will be printed.

maxiter

Maximum number of iterations.

Details

It is assumed that the data is a random sample from the model x = m + A s where the latent vector s = (s_1^T, s_2^T)^T consists of k-dimensional non-Gaussian subvector (the signal) and p - k-dimensional Gaussian subvector (the noise) and the components of s are mutually independent. Without loss of generality we further assume that the components of s have zero means and unit variances.

To test the null hypothesis H_0: k_{true} \leq k the algorithm first estimates k + 1 components using delfation-based NGPP with the chosen non-linearities and weighting. Under the null hypothesis the distribution of the final p - k components is standard multivariate normal and the significance of the test is obtained by comparing the objective function value of the (k + 1)th estimated components to the same quantity estimated from N samples of size n from (p - k)-dimensional standard multivariate normal distribution.

Note that if maxiter is reached at any step of the algorithm it will use the current estimated direction and continue to the next step.

Value

A list with class 'ictest', inheriting from the class 'hctest', containing the following components:

statistic

Test statistic, i.e. the objective function value of the (k + 1)th estimated component.

p.value

Obtained p-value.

parameter

Number N of simulated normal samples.

method

Character string denoting which test was performed.

data.name

Character string giving the name of the data.

alternative

Alternative hypothesis, i.e. "There are less than p - k Gaussian components".

k

Tested dimension k.

W

Estimated unmixing matrix

S

Matrix of size n \times (k + 1) containing the estimated signals.

D

Vector of the objective function values of the signals

MU

Location vector of the data which was substracted before estimating the signal components.

Author(s)

Joni Virta

References

Virta, J., Nordhausen, K. and Oja, H., (2016), Projection Pursuit for non-Gaussian Independent Components, <https://arxiv.org/abs/1612.05445>.

See Also

NGPP, NGPPest

Examples

# Simulated data with 2 signals and 2 noise components

n <- 200
S <- cbind(rexp(n), rbeta(n, 1, 2), rnorm(n), rnorm(n))
A <- matrix(rnorm(16), ncol = 4)
X <- S %*% t(A)

# The number of simulations N should be increased in practical situations
# Now we settle for N = 100

res1 <- NGPPsim(X, 1, N = 100)
res1
screeplot(res1)

res2 <- NGPPsim(X, 2, N = 100)
res2
screeplot(res2)