Description Usage Arguments Details Value Author(s) References See Also Examples

The FixNA (Shi et al., 2009) and FixNA2 (Matilainen et al., 2017) methods for blind source separation problem. It is used for time series with stochastic volatility. These methods are alternatives to vSOBI method.

1 2 3 4 5 6 |

`X` |
A numeric matrix or a multivariate time series object of class |

`k` |
A vector of lags. It can be any non-zero positive integer, or a vector consisting of them. Default is |

`eps` |
Convergence tolerance. |

`maxiter` |
The maximum number of iterations. |

`G` |
Function |

`method` |
The method to be used. The choices are |

`...` |
Further arguments to be passed to or from methods. |

Assume that a *p*-variate *Y* with *T* observations is whitened, i.e. *Y = S^(-1/2)*(X_t - (1/T)*sum_t(X_t))*, for *t = 1, …, T*,
where *S* is the sample covariance matrix of *X*. The algorithm for method FixNA finds an orthogonal matrix *U* by maximizing

*D_1(U) = sum_k(D_k1(U)) = sum_i(sum_k((1/(T - k))*sum_t[G(u_i' Y_t)*G(u_i' Y_(t + k))])) *

and the algorithm for method FixNA2 finds an orthogonal matrix *U* by maximizing

*D_2(U) = sum_k(D_k2(U))*

*= sum_k(sum_i(|(1/(T - k))*sum_t[G(u_i' Y_t)*G(u_i' Y_(t + k))] - (1/(T - k))^2*sum_t[G(u_i' Y_t)]*sum_t[G(u_i' Y_(t + k))]|)),*

where *i = 1, …, p*, *k = 1, …, K* and *t = 1, …, 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 *U = (u_1, u_2, …, u_p)'*.

Matrix *T_mik* is a partial derivative of *D_mk(U)*, for *m = 1, 2*, with respect to *u_i*.
Then *T_mk = (T_m1k, …, T_mpk)'*, where *p* is the number of columns in *Y*, and *T = sum(T_mk)*.
The update for the orthogonal matrix *U.new = (T_m T_m')^(-1/2)*T_m* is calculated at each iteration step. The algorithm stops when

*||U.new - U.old||*

is less than `eps`

.
The final unmixing matrix is then *W = U S^(-1/2)*.

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

`W ` |
The estimated unmixing matrix. |

`k ` |
The vector of the used lags. |

`S ` |
The estimated sources as time series object standardized to have mean 0 and unit variances. |

Markus Matilainen

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.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 | ```
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)$y
# create a daily time series
X <- ts(cbind(s1, s2, s3) %*% t(A), end = c(2015, 338), frequency = 365.25)
res <- FixNA(X)
res
coef(res)
plot(res)
head(bss.components(res))
MD(res$W, A) # Minimum Distance Index, should be close to zero
``` |

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