A-short-introduction-to-tensorFun

1 Quick start

We go through some basic functions in ‘tensorFun’ in this vignettes. First, we start with constructing a tensor, which is by using the base class ‘array’ in R. The following tensor is an order-\(3\) tensor with dimensions \(3\times 4\times 2\).

a <- array(1:24, dim=c(3,4,2))

As we can see, subsetting and truncating in multi-dimensional arrays are trivial, and we display them below by an example to inject missingness inside the tensor.

a[3,1,1] <- NA
print(a)
#> , , 1
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,]    1    4    7   10
#> [2,]    2    5    8   11
#> [3,]   NA    6    9   12
#> 
#> , , 2
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,]   13   16   19   22
#> [2,]   14   17   20   23
#> [3,]   15   18   21   24

Now we perform rearrangement for the tensor, and the result displayed shows the validity of the algorithm. The following is subsequent rearrangement along mode \(1\), mode \(2\), and mode \(3\), respectively.

b <- a
b <- rearrange(b, 1)$ten
b <- rearrange(b, 2)$ten
b <- rearrange(b, 3)$ten
print(b)
#> , , 1
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,]   16   19   22   13
#> [2,]   17   20   23   14
#> [3,]   18   21   24   15
#> 
#> , , 2
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,]    4    7   10    1
#> [2,]    5    8   11    2
#> [3,]    6    9   12   NA

Indeed, the reverse algorithm for rearrangement returns back to the original tensor we have, but we need to keep track of the arranged set of indices along each mode.

# rearrange
c <- rearrange(a, 1)
l1 <- c$l
c <- rearrange(c$ten, 2)
l2 <- c$l
c <- rearrange(c$ten, 3)
l3 <- c$l

# rearrange backwards
d <- rearrange_inv(c$ten, 1, l1)
d <- rearrange_inv(d, 2, l2)
d <- rearrange_inv(d, 3, l3)
print(d==a)
#> , , 1
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,] TRUE TRUE TRUE TRUE
#> [2,] TRUE TRUE TRUE TRUE
#> [3,]   NA TRUE TRUE TRUE
#> 
#> , , 2
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,] TRUE TRUE TRUE TRUE
#> [2,] TRUE TRUE TRUE TRUE
#> [3,] TRUE TRUE TRUE TRUE

\(\textbf{Notice}:\) The mode order of ‘rearrange’ and ‘rearrange_inv’ does NOT matter, but more caution is needed in storing the rearrangement list of indices ‘rearrange’. For simplicity, see section 2 for a one-click rearrangement algorithm facilitating sweeping through rearrangement algorithm along all modes.

2 Rearrangement algorithm

The rearrangement algorithm was proposed by Bai and Ng (Jushan Bai 2021) on vector time series, and hence the entire data is a matrix. The rearrangement algorithm could be extended to high order tensor data, as shown in section 1, with order-2 tensor returning back to the algorithm of Bai and Ng.

Ideally, we might often want to sweep through all modes of the tensor data. If so, instead of performing rearrangement algorithm on each mode, we provide the ‘All_rearrange’ function to rearrange the data entirely.

All_rearrange(a)
#> $tensor
#> , , 1
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,]   16   19   22   13
#> [2,]   17   20   23   14
#> [3,]   18   21   24   15
#> 
#> , , 2
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,]    4    7   10    1
#> [2,]    5    8   11    2
#> [3,]    6    9   12   NA
#> 
#> 
#> $ind
#> $ind[[1]]
#> [1] 3
#> 
#> $ind[[2]]
#> [1] 1
#> 
#> $ind[[3]]
#> [1] 1

Notice all modes including the time mode should be rearranged. Furthermore, in case some modes are not to be rearranged, we can state the excluded modes in the parameter space.

Similarly, we provide the ‘All_inv’ function to reverse the rearrangement for the entire data, and the reversed data shall be the original one, as shown below.

All_inv(All_rearrange(a)) == a
#> , , 1
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,] TRUE TRUE TRUE TRUE
#> [2,] TRUE TRUE TRUE TRUE
#> [3,]   NA TRUE TRUE TRUE
#> 
#> , , 2
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,] TRUE TRUE TRUE TRUE
#> [2,] TRUE TRUE TRUE TRUE
#> [3,] TRUE TRUE TRUE TRUE

3 Unfolding, refolding, and k-mode matrix product

We now quickly go through the basic unfolding and refolding functions as an example.

a.1 <- unfold(a, 1)
print(a.1)
#>      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
#> [1,]    1    4    7   10   13   16   19   22
#> [2,]    2    5    8   11   14   17   20   23
#> [3,]   NA    6    9   12   15   18   21   24

Without doubt, the refolding of an unfolding matrix returns back to the original tensor, given the correct dimension.

refold(a.1, 1, dim(a))==a
#> , , 1
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,] TRUE TRUE TRUE TRUE
#> [2,] TRUE TRUE TRUE TRUE
#> [3,]   NA TRUE TRUE TRUE
#> 
#> , , 2
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,] TRUE TRUE TRUE TRUE
#> [2,] TRUE TRUE TRUE TRUE
#> [3,] TRUE TRUE TRUE TRUE

Lastly, k-mode matrix product is performed.

ttm(a, matrix(1:4,nrow=2), 3)
#> , , 1
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,]   40   52   64   76
#> [2,]   44   56   68   80
#> [3,]   NA   60   72   84
#> 
#> , , 2
#> 
#>      [,1] [,2] [,3] [,4]
#> [1,]   54   72   90  108
#> [2,]   60   78   96  114
#> [3,]   NA   84  102  120

4 Supplementary

Some examples are given here for ‘obs_ind’. It provides the largest index without missing entries along a specified mode. For example, tensor \(a\) has a missing entry at the index \([3,1,1]\), and hence the largest observed index for mode \((1,2,3)\) would be \([2,0,0]\).

print(obs_ind(a,1))
#> [1] 2
print(obs_ind(a,2))
#> [1] 0
print(obs_ind(a,3))
#> [1] 0

More explanations for other functions will be given, once all works are done.

A-short-introduction-to-tensorFun

1 Quick start

2 Rearrangement algorithm

3 Unfolding, refolding, and k-mode matrix product

4 Supplementary

References