SLIDE 1
Martin Nilsson Jacobi
Identification of weak lumpability in Markov chains
General criteria for weak lumpability found, and the structure of the corresponding projection
- perator is derived
Identification of weak lumpability in Markov chains General - - PowerPoint PPT Presentation
Identification of weak lumpability in Markov chains General - - PowerPoint PPT Presentation
Identification of weak lumpability in Markov chains General criteria for weak lumpability found, and the structure of the corresponding projection operator is derived Martin Nilsson Jacobi Projections and the Markov property e e P P
SLIDE 2
SLIDE 3
markov lumping (aggregation)
aggregates
variables/states
π = 1 1 1 1 e P π π P
Aggregation of state 2 and 3 at the micro level into one state at the macro level
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e P
ρi e PKL = X
i∈L
ρi P
m∈L ρm
X
j∈K
Pj←i e P = πPπ+ π+ = D1/2 ⇣ πD1/2⌘† A† =
- AT A
−1 AT Dii = ρi e P π π P
SLIDE 5
π π P e P π P e P st st+1 st+2 ˜ st ˜ st+1 ˜ st+2 e P N = ⇢ πP Nπ+ (πPπ+)N πP Nπ+ =
- πPπ+N
p(˜ st+2|˜ st+1˜ st˜ st−1 · · · ) independent of ˜ st˜ st−1 · · · . ∀N e P N
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πPπ+πPπ+ = πPPπ+ π π P e P
simple weak lumpability
Pπ+ = π+ e P e P π π P
strong lumpability
e Pπ = πP πPπ+π = πP π+πPπ+ = Pπ+
Either of these eq. are sufficient but not necessary for weak lumpability
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Invariance conditions
Interpretation of the conditions
π+ e P = Pπ+
Weak lumping lumping
Column space of π+ invariant under P
e Pπ = πP
Strong lumping
Row space of π invariant under P T
Invariance typically means eigenvactors
πP Nπ+ =
- πPπ+N
These commutation relations also solve
SLIDE 8
strong lumping
xt+1 = Pxt
Π = 1 1 1 1
aggregates
variables/states
Row-space spanned by eigenvectors of PT
- a
b b c ⇥
Linear combination of the rows:
Search for eigenvectors with constant level structure!
#levels = #aggregates = #eigenvectors with that level structure
y = Πx
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Weak lumping
π+ = 1
ρ2 ρ2+ρ3 ρ3 ρ2+ρ3
1 Take right eigenvectors of P, say u. Look for level structure in the vector vi = ui/ρi.
Column-space spanned by eigenvectors of P
#levels = #aggregates = #eigenvectors with that level structure
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Example
P = 0.25 0. 0.875 0.25 0.166667 0.125 0.5 0.833333 0.
(−0.721995, −0.309426, −0.618853) (−0.801784, 0.267261, 0.534522) (0.813733, −0.348743, −0.464991)
Right eig vec Left eig vec
(−0.57735, −0.57735, −0.57735) (−0.0733017, −0.855186, 0.513112) (0.0000, −0.894427, 0.447214) (1., 1., 1.) (1.11051, −0.863731, −0.863731) (−1.12706, 1.12706, 0.751375)
/ρi
No strong lumping Weak lumping: {{1},{2,3}}
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The general case
πP Nπ+ =
- πPπ+N
∀N
Assume a projection to n lumps. π gives a weak lumping iff
- 1. the row-space of π is spanned by left eigenvectors with index i1, i2, . . . , ik
- 2. the column-space of π+ is spanned by right eigenvectors with index j1, j2, . . . , jm
- 3. the number of indices in common between π and π+, |{i1, i2, . . . , ik} ∩ {j1, j2, . . . , jm}|,
is exactly equal to n.
SLIDE 12
So we have derived the general structure of a projection that gives weak lumpability in the most general form. (Good.)
- The general criterion is not constructive in the