1 1 1/3 2 1/3 3 3 2/3 1/3 0 1 0 P 1/3 0 2/3 1/3 1/3 1/3
1/3 1/3 1/3 1/3 2/3 1 1 2 3 P = ⇤ 010 1/302/3 1/3 1/3 1/3 ⇥ ⌅
Convergence 0 1 0 1/3 P= 1/3 0 2/3 /3 1/3 1/3 1/3 0.2860 0.0a 0.6660 He ≈ 0.2500 0.3630 0.3888 0.2308 0.6860 0.3830 V distribution m, πP20≈(8，)
Convergence P = ⇤ 010 1/302/3 1/3 1/3 1/3 ⇥ ⌅ 1/3 1/3 1/3 1/3 2/3 1 1 2 3 P5 ⇤ 0.2469 0.4074 0.3457 0.2510 0.3621 0.3868 0.2510 0.3663 0.3827 ⇥ P10 ⌅ ⇤ 0.2500 0.3747 0.3752 0.2500 0.3751 0.3749 0.2500 0.3751 0.3749 ⇥ P20 ⌅ ⇤ 0.2500 0.3750 0.3750 0.2500 0.3750 0.3750 0.2500 0.3750 0.3750 ⇥ ⌅ distribution , P20 ( 1 4 , 3 8 , 3 8 ) P2 ⇤ 0.3333 0 0.6667 0.3333 0.5556 0.2222 0.2778 0.6111 0.3333 ⇥ ⌅
Stationary Distribution Markoy chain=(,P) stationary distribution πP=π (fixed point) Perron-Frobenius Theorem: stochastic matrix P:P1 =1 1 is also a left eigenvalue of P (eigenvalue of pT) ●the left eigenvectorπP=πis nonnegative stationary distribution always exists
Stationary Distribution • stationary distribution π: • Perron-Frobenius Theorem: • stochastic matrix P: • 1 is also a left eigenvalue of P (eigenvalue of PT) • the left eigenvector is nonnegative • stationary distribution always exists Markov chain M = (⌦, P) ⇡P = ⇡ P1 = 1 ⇡P = ⇡ (fixed point)
Perron-Frobenius Perron-Frobenius Theorem: A:a nonnegative nxn matrix with spectral radius (A) (A)>0 is an eigenvalue of A; there is a nonnegative (left and right)eigenvector associated with o(A); if further A is irreducible,then: there is a positive (left and right)eigenvector associated with o(A)that is of multiplicity 1; for stochastic matrix A the spectral radius o(A)=1
Perron-Frobenius • A : a nonnegative n×n matrix with spectral radius ρ(A) • ρ(A) > 0 is an eigenvalue of A; • there is a nonnegative (left and right) eigenvector associated with ρ(A); • if further A is irreducible, then: • there is a positive (left and right) eigenvector associated with ρ(A) that is of multiplicity 1; • for stochastic matrix A the spectral radius ρ(A)=1. Perron-Frobenius Theorem:
Stationary Distribution Markoy chain=(,P) stationary distribution πP=π (fixed point) Perron-Frobenius Theorem: stochastic matrix P:P1 =1 1 is also a left eigenvalue of P (eigenvalue of pT) ●the left eigenvectorπP=πis nonnegative stationary distribution always exists
Stationary Distribution • stationary distribution π: • Perron-Frobenius Theorem: • stochastic matrix P: • 1 is also a left eigenvalue of P (eigenvalue of PT) • the left eigenvector is nonnegative • stationary distribution always exists Markov chain M = (⌦, P) ⇡P = ⇡ P1 = 1 ⇡P = ⇡ (fixed point)