FixNA {tsBSS}

The FixNA Method for Blind Source Separation

Description

The FixNA (Fixed-point algorithm for maximizing the Nonlinear Autocorrelation; Shi et al., 2009) and FixNA2 (Matilainen et al., 2017) methods for blind source separation of time series with stochastic volatility. These methods are alternatives to vSOBI method.

Usage

FixNA(X, ...)

## Default S3 method:
FixNA(X, k = 1:12, eps = 1e-06, maxiter = 1000, G = c("pow", "lcosh"),
      method = c("FixNA", "FixNA2"),
      ordered = FALSE, acfk = NULL, original = TRUE, alpha = 0.05, ...)
## S3 method for class 'ts'
FixNA(X, ...)
## S3 method for class 'xts'
FixNA(X, ...)
## S3 method for class 'zoo'
FixNA(X, ...)

Arguments

X

A numeric matrix or a multivariate time series object of class ts, xts or zoo. Missing values are not allowed.

k

A vector of lags. It can be any non-zero positive integer, or a vector consisting of them. Default is 1:12.

eps

Convergence tolerance.

maxiter

The maximum number of iterations.

G

Function G(x). The choices are "pow" (default) and "lcosh".

method

The method to be used. The choices are "FixNA" (default) and "FixNA2".

ordered

Whether to order components according to their volatility. Default is FALSE.

acfk

A vector of lags to be used in testing the presence of serial autocorrelation. Applicable only if ordered = TRUE.

original

Whether to return the original components or their residuals based on ARMA fit. Default is TRUE, i.e. the original components are returned. Applicable only if ordered = TRUE.

alpha

Alpha level for linear correlation detection. Default is 0.05.

...

Further arguments to be passed to or from methods.

Details

Assume that a p-variate {\bf Y} with T observations is whitened, i.e. {\bf Y}={\bf S}^{-1/2}({\bf X}_t - \frac{1}{T}\sum_{t=1}^T {\bf X}_{t}), for t = 1, \ldots, T, where {\bf S} is the sample covariance matrix of {\bf X}. The algorithm for method FixNA finds an orthogonal matrix {\bf U} by maximizing

{\bf D}_1({\bf U}) = \sum_{k = 1}^K {\bf D}_{1k}({\bf U})= \sum_{k = 1}^K \sum_{i = 1}^p \frac{1}{T - k}\sum_{t=1}^{T - k}[G({\bf u}_i' {\bf Y}_t) G({\bf u}_i' {\bf Y}_{t + k})]

and the algorithm for method FixNA2 finds an orthogonal matrix {\bf U} by maximizing

{\bf D}_2({\bf U}) = \sum_{k = 1}^K {\bf D}_{2k}({\bf U})

= \sum_{k = 1}^K \sum_{i = 1}^p\left|\frac{1}{T - k}\sum_{t=1}^{T - k}[G({\bf u}_i' {\bf Y}_t) G({\bf u}_i' {\bf Y}_{t + k})] - \left(\frac{1}{T - k}\right)^2\sum_{t=1}^{T - k}[G({\bf u}_i' {\bf Y}_t)]\sum_{t=1}^{T - k}[G({\bf u}_i' {\bf Y}_{t + k})]\right|.

where i = 1, \ldots, p, k = 1, \ldots, K and t = 1, \ldots, T. For function G(x) the choices are x^2 and log(cosh(x)).

The algorithm works iteratively starting with diag(p) as an initial value for an orthogonal matrix {\bf U} = ({\bf u}_1, {\bf u}_2, \ldots, {\bf u}_p)'.

Matrix {\bf T}_{mik} is a partial derivative of {\bf D}_{mk}({\bf U}), for m = 1, 2, with respect to {\bf u}_i. Then {\bf T}_{mk} = ({\bf T}_{m1k}, \ldots, {\bf T}_{mpk})', where p is the number of columns in {\bf Y}, and {\bf T}_m = \sum_{k = 1}^K {\bf T}_{mk}. The update for the orthogonal matrix {\bf U}_{new} = ({\bf T}_m{\bf T}_m')^{-1/2}{\bf T}_m is calculated at each iteration step. The algorithm stops when

||{\bf U}_{new} - {\bf U}_{old}||

is less than eps. The final unmixing matrix is then {\bf W} = {\bf US}^{-1/2}.

For ordered = TRUE the function orders the sources according to their volatility. First a possible linear autocorrelation is removed using auto.arima. Then a squared autocorrelation test is performed for the sources (or for their residuals, when linear correlation is present). The sources are then put in a decreasing order according to the value of the test statistic of the squared autocorrelation test. For more information, see lbtest.

Value

A list of class 'bssvol', inheriting from class 'bss', containing the following components:

W

The estimated unmixing matrix. If ordered = TRUE, the rows are ordered according to the order of the components.

k

The vector of the used lags.

S

The estimated sources as time series object standardized to have mean 0 and unit variances. If ordered = TRUE, then components are ordered according to their volatility. If original = FALSE, the sources with linear autocorrelation are replaced by their ARMA residuals.

MU

The mean vector of X.

If ordered = TRUE, then also the following components included in the list:

Sraw

The ordered original estimated sources as time series object standardized to have mean 0 and unit variances. Returned only if original = FALSE.

fits

The ARMA fits for the components with linear autocorrelation.

armaeff

A logical vector. Is TRUE if ARMA fit was done to the corresponding component.

linTS

The value of the modified Ljung-Box test statistic for each component.

linP

p-value based on the modified Ljung-Box test statistic for each component.

volTS

The value of the volatility clustering test statistic.

volP

p-value based on the volatility clustering test statistic.

Author(s)

Markus Matilainen

References

Hyvärinen, A. (2001), Blind Source Separation by Nonstationarity of Variance: A Cumulant-Based Approach, IEEE Transactions on Neural Networks, 12(6): 1471–1474.

Matilainen, M., Miettinen, J., Nordhausen, K., Oja, H. and Taskinen, S. (2017), On Independent Component Analysis with Stochastic Volatility Models, Austrian Journal of Statistics, 46(3–4), 57–66.

Shi, Z., Jiang, Z. and Zhou, F. (2009), Blind Source Separation with Nonlinear Autocorrelation and Non-Gaussianity, Journal of Computational and Applied Mathematics, 223(1): 908–915.

See Also

vSOBI, lbtest, auto.arima

Examples

if(require("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

# Create a daily time series
X <- ts(cbind(s1, s2, s3) %*% t(A), end = c(2015, 338), frequency = 365.25)

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

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