Generalized JADE

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Description

The gJADE method for blind source separation problem. It is designed for time series with stochastic volatility. The method is a generalization of JADE, which is a method for blind source separation problem using only marginal information.

Usage

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gJADE(X, ...)

## Default S3 method:
gJADE(X, k = 0:12, eps = 1e-06, maxiter = 100, method = "frjd", ...)
## S3 method for class 'ts'
gJADE(X, ...)

Arguments

X

Numeric matrix or multivariate time series object of class ts. Missing values are not allowed.

k

Vector of lags. Lag can be any non-negative integer, or a vector consisting of them. Default is 0:12. If k = 0, this method reduces to JADE.

eps

Convergence tolerance.

maxiter

Maximum number of iterations.

method

Method to use for the joint diagonalization, options are rjd and frjd. Default is frjd.

...

Other arguments passed on to chosen joint diagonalization method.

Details

Assume that Y has p columns and it is whitened, i.e. Y = S^(-1/2)*(X - (1/T)*sum_t(X_(ti))), where S is a sample covariance matrix of X. The matrix C^ij_k(Y) is of the form

C^ij_k(Y) = B^ij_k(Y) - S_k(Y) (E^ij + E^ji) S_k(Y)' - trace(E^ij)*I,

for i, j = 1, …, p, where S_k(Y) is a lagged sample covariance matrix of Y, E^ij is a matrix where element (i,j) equals to 1 and all other elements are 0, I is an identity matrix of order p and B^ij_k(Y) is as in gFOBI.

The algorithm finds an orthogonal matrix U by maximizing

sum_i(sum_j (sum_k (||diag(U C^ij_k(Y) U')||^2))),

where k = 1, …, K. The final unmixing matrix is then W = U S^(-1/2).

Value

A list with class 'bss' containing the following components:

W

The estimated unmixing matrix.

k

The vector of the used lags.

S

Estimated sources as time series object standardized to have mean 0 and unit variances.

Author(s)

Klaus Nordhausen

References

Cardoso, J.-F., Souloumiac, A., (1993). Blind beamforming for non-Gaussian signals, in: IEE-Proceedings-F, volume 140, pp. 362–370.

Matilainen, M., Nordhausen, K. and Oja, H. (2015), New independent component analysis tools for time series, Statistics & Probability Letters, 105, 80–87.

See Also

frjd, JADE, gFOBI

Examples

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library(stochvol)
n <- 10000
A <- matrix(rnorm(9), 3, 3)

# simulate SV models
s1 <- svsim(n, mu = -10, phi = 0.8, sigma = 0.1)$y
s2 <- svsim(n, mu = -10, phi = 0.9, sigma = 0.2)$y
s3 <- svsim(n, mu = -10, phi = 0.95, sigma = 0.4)$y

X <- cbind(s1, s2, s3) %*% t(A)

res <- gJADE(X)
res
coef(res)
plot(res)
head(bss.components(res))

MD(res$W, A) # Minimum Distance Index, should be close to zero