Temporal IFIMs - Index Matrices: Towards an Augmented Matrix Calculus - page 67

Chapter 4
Temporal IFIMs

We introduce the definition of the object Temporal IFIM (TIFIM), described in the
paper [25] of Evdokia Sotirova, Veselina Bureva, Anthony Shannon and the author,
by

A (T ) =[ K , L , T , { μ k i , l j ,τ , ν k i , l j ,τ }]

⎧
⎨

⎫
⎬

l 1 l 2 ... l n

k 1 μ k 1 , l 1 ,τ , ν k 1 , l 1 ,τ μ k 1 , l 2 ,τ , ν k 1 , l 2 ,τ ... μ k 1 , l n ,τ , ν k 1 , l n ,τ

. . . .

. . .

≡ | τ ∈ T ,

μ k i , l 1 ,τ , ν k i , l 1 ,τ μ k i , l 2 ,τ , ν k i , l 2 ,τ ... μ k i , l n ,τ , ν k i , l n ,τ

⎩ k i ⎭

. . . .

. . .

k m μ k m , l 1 ,τ , ν k m , l 1 ,τ μ k m , l 2 ,τ , ν k m , l 2 ,τ ... μ k m , l n ,τ , ν k m , l n ,τ

where for every τ ∈ T ,1 ≤ i ≤ m , 1 ≤ j ≤ n :

μ k i , l j ,τ , ν k i , l j ,τ , μ k i , l j ,τ + ν k i , l j ,τ ∈[ 0 , 1 ] .

is its element, i.e., a time-moment.
By analogy with the Chap. 2 , we extend the concept of a TIFIM, defining an
Extended TIFIM (ETIFIM; see [20]), by:

Here, T is a some fixed temporal scale and τ

A ∗ (T ) =[ K ∗ (T ), L ∗ (T ), { μ k i , l j ,τ , ν k i , l j ,τ }]