tNSS.SD {tensorBSS}

NSS-SD Method for Tensor-Valued Time Series

Description

Estimates the non-stationary sources of a tensor-valued time series using separation information contained in two time intervals.

Usage

tNSS.SD(x, n.cuts = NULL)

Arguments

x

Numeric array of an order at least two. It is assumed that the last dimension corresponds to the sampling units.

n.cuts

Either a 3-vector of interval cutoffs (the cutoffs are used to define the two intervals that are open below and closed above, e.g. (a, b]) or NULL (the time range is sliced into two parts of equal size).

Details

Assume that the observed tensor-valued time series comes from a tensorial BSS model where the sources have constant means over time but the component variances change in time. Then TNSS-SD estimates the non-stationary sources by dividing the time scale into two intervals and jointly diagonalizing the covariance matrices of the two intervals within each mode.

Value

A list with class 'tbss', inheriting from class 'bss', containing the following components:

S

Array of the same size as x containing the independent components.

W

List containing all the unmixing matrices.

EV

Eigenvalues obtained from the joint diagonalization.

n.cuts

The interval cutoffs.

Xmu

The data location.

datatype

Character string with value "ts". Relevant for plot.tbss.

Author(s)

Joni Virta

References

Virta J., Nordhausen K. (2017): Blind source separation for nonstationary tensor-valued time series, 2017 IEEE 27th International Workshop on Machine Learning for Signal Processing (MLSP), doi: 10.1109/MLSP.2017.8168122

See Also

NSS.SD, NSS.JD, NSS.TD.JD, tNSS.JD, tNSS.TD.JD

Examples

# Create innovation series with block-wise changing variances

# 9 smooth variance structures
var_1 <- function(n){
  t <- 1:n
  return(1 + cos((2*pi*t)/n)*sin((2*150*t)/(n*pi)))
}

var_2 <- function(n){
  t <- 1:n
  return(1 + sin((2*pi*t)/n)*cos((2*150*t)/(n*pi)))
}

var_3 <- function(n){
  t <- 1:n
  return(0.5 + 8*exp((n+1)^2/(4*t*(t - n - 1))))
}

var_4 <- function(n){
  t <- 1:n
  return(3.443 - 8*exp((n+1)^2/(4*t*(t - n - 1))))
}

var_5 <- function(n){
  t <- 1:n
  return(0.5 + 0.5*gamma(10)/(gamma(7)*gamma(3))*(t/(n + 1))^(7 - 1)*(1 - t/(n + 1))^(3 - 1))
}

var_6 <- function(n){
  t <- 1:n
  res <- var_5(n)
  return(rev(res))
}

var_7 <- function(n){
  t <- 1:n
  return(0.2+2*t/(n + 1))
}

var_8 <- function(n){
  t <- 1:n
  return(0.2+2*(n + 1 - t)/(n + 1))
}

var_9 <- function(n){
  t <- 1:n
  return(1.5 + cos(4*pi*t/n))
}


# Innovation series
n <- 1000

innov1 <- c(rnorm(n, 0, sqrt(var_1(n))))
innov2 <- c(rnorm(n, 0, sqrt(var_2(n))))
innov3 <- c(rnorm(n, 0, sqrt(var_3(n))))
innov4 <- c(rnorm(n, 0, sqrt(var_4(n))))
innov5 <- c(rnorm(n, 0, sqrt(var_5(n))))
innov6 <- c(rnorm(n, 0, sqrt(var_6(n))))
innov7 <- c(rnorm(n, 0, sqrt(var_7(n))))
innov8 <- c(rnorm(n, 0, sqrt(var_8(n))))
innov9 <- c(rnorm(n, 0, sqrt(var_9(n))))

# Generate the observations
vecx <- cbind(as.vector(arima.sim(n = n, list(ar = 0.9), innov = innov1)),
              as.vector(arima.sim(n = n, list(ar = c(0, 0.2, 0.1, -0.1, 0.7)), 
              innov = innov2)),
              as.vector(arima.sim(n = n, list(ar = c(0.5, 0.3, -0.2, 0.1)), 
              innov = innov3)),
              as.vector(arima.sim(n = n, list(ma = -0.5), innov = innov4)),
              as.vector(arima.sim(n = n, list(ma = c(0.1, 0.1, 0.3, 0.5, 0.8)), 
              innov = innov5)),
              as.vector(arima.sim(n = n, list(ma = c(0.5, -0.5, 0.5)), innov = innov6)),
              as.vector(arima.sim(n = n, list(ar = c(-0.5, -0.3), ma = c(-0.2, 0.1)), 
              innov = innov7)),
              as.vector(arima.sim(n = n, list(ar = c(0, -0.1, -0.2, 0.5), ma = c(0, 0.1, 0.1, 0.6)),
              innov = innov8)),
              as.vector(arima.sim(n = n, list(ar = c(0.8), ma = c(0.7, 0.6, 0.5, 0.1)),
              innov = innov9)))


# Vector to tensor
tenx <- t(vecx)
dim(tenx) <- c(3, 3, n)


# Run TNSS-SD
res <- tNSS.SD(tenx)
res$W