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I would like some feedback regarding this process or the meaning of this process.

Let say that I have a discrete time series:
S = [1 2 3 4 5]

And that I represent this serie by a stochastic matrix M (5x3):
0 0 1
0 1 0
1 0 0
0 0 1
0 1 0

This stochastic matrix come from the Matrix N (6x3), such that
M(1,1:3) = N(2,1:3) - N(1,2:3)
M(2,1:3) = N(3,1:3) - N(2,2:3)
M(3,1:3) = N(4,1:3) - N(3,2:3)
M(4,1:3) = N(5,1:3) - N(4,2:3)
M(5,1:3) = N(6,1:3) - N(5,2:3)

With N:
0 0 0
0 0 1
0 1 1
1 1 1
1 1 2
1 2 2

With Sum of Columns:
3 5 7

Now if you multiply S x M, the resulting vector is:
3 7 5

Note that this does not work for all matrices!
e.g.
0 0 1
0 0 1
0 1 0
1 0 0
0 0 1

from
0 0 0
0 0 1
0 0 2
0 1 2
1 1 2
1 1 3

With Sum of Columns:
2 3 10

And if you multiply S x M, the resulting vector is:
4 3 8

So, if some of you could give me informations regarding the meaning of this process, if it make sense, it it mean something in mathematics!

Thanks in advance

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