# The m-Mode Autocovariance Matrix

### Description

Estimates the m-mode autocovariance matrix from an array of array-valued observations with the specified lag.

### Usage

1 | ```
mModeAutoCovariance(x, m, lag, center = TRUE)
``` |

### Arguments

`x` |
Array of order higher than two with the last dimension corresponding to the sampling units. |

`m` |
The mode with respect to which the autocovariance matrix is to be computed. |

`lag` |
The lag with respect to which the autocovariance matrix is to be computed. |

`center` |
Logical, indicating whether the observations should be centered prior to computing the autocovariance matrix. Default is |

### Details

The m-mode autocovariance matrix provides a higher order analogy for the ordinary autocovariance matrix of a random vector and is computed for a random tensor *X_t* of size *p_1 x p_2 x ... x p_r* as *Cov_m,tau(X_t) = E(X_t(m) X_(t+tau)(m)^T)/(p_1 ... p_(m-1) p_(m+1) ... p_r)*, where *X_t(m)* is the centered *m*-flattening of *X_t* and *tau* is the desired `lag`

. The algorithm computes the estimate of this based on the sample `x`

.

### Value

The `m`

-mode autocovariance matrix of `x`

with respect to `lag`

having the size *p_m x p_m*.

### Author(s)

Joni Virta

### References

Virta, J. and Nordhausen, K., (2016), Blind source separation of tensor-valued time series, *submitted*, **???**, ???–???.

### See Also

`mModeCovariance`

### Examples

1 2 3 4 5 6 7 8 9 10 11 | ```
n <- 1000
S <- t(cbind(as.vector(arima.sim(n = n, list(ar = 0.9))),
as.vector(arima.sim(n = n, list(ar = -0.9))),
as.vector(arima.sim(n = n, list(ma = c(0.5, -0.5)))),
as.vector(arima.sim(n = n, list(ar = c(-0.5, -0.3)))),
as.vector(arima.sim(n = n, list(ar = c(0.5, -0.3, 0.1, -0.1), ma=c(0.7, -0.3)))),
as.vector(arima.sim(n = n, list(ar = c(-0.7, 0.1), ma = c(0.9, 0.3, 0.1, -0.1))))))
dim(S) <- c(3, 2, n)
mModeAutoCovariance(S, m = 1, lag = 1)
mModeAutoCovariance(S, m = 1, lag = 4)
``` |

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