Kano Lab. Yuchi MATSUOKA December 22, 2016 1 / 32
1 1.1 1.2 1.3 1.4 2 ARMA 2.1 ARMA 2 / 32
1 1.1 1.2 1.3 1.4 2 ARMA 2.1 ARMA 3 / 32
1.1.1 - - - 4 / 32
1.1.2 - - - - - 5 / 32
1.1.3 y t µ t = E(y t ), V ar(y t ) = E(y t µ t ) 2. V ar(y t ) γ kt = Cov(y t, y t k ) = E[(y t µ t )(y t k µ t k ]. - - k 6 / 32
ρ kt = Corr(y t, y t k ) = Cov(y t, y t k ) V ar(yt ) V ar(y t k ) = γ kt γ0t γ 0,t k - - ρ 0t = 1. - k - 7 / 32
t t {y t } T t=1 {y t} t= t= 8 / 32
1 1.1 1.2 1.3 1.4 2 ARMA 2.1 ARMA 9 / 32
- Definition ( ) t k E(y t ) = µ Cov(y t, y t k ) = E[(y t µ)(y t k µ)] = γ k. 10 / 32
Definition ( ) t k (y t, y t+1,..., y t+k ) T i.e, t,k 11 / 32
1 1.1 1.2 1.3 1.4 2 ARMA 2.1 ARMA 12 / 32
Definition (iid ) i.i.d. iid iid 0 iid Definition ( ) t σ 2, k = 0 E(ϵ t ) = 0, γ k = E(ϵ t ϵ t k ) = 0, k 0 ϵ t 13 / 32
1 1.1 1.2 1.3 1.4 2 ARMA 2.1 ARMA 14 / 32
1 ȳ = 1 T T y t, ˆγ k = 1 T t=1 T t=k+1 (y t ȳ)(y t k ȳ), k = 0, 1, 2,... ˆρ k = ˆγ k ˆγ 0, k = 1, 2, 3,... 2 H 0 : ρ k = 0 vs H 1 0 3 ρ k N(0, 1/T ) 15 / 32
H 0 : ρ 1 = ρ 2 = = ρ m = 0 vs H 1 : k [1, m] ρ k 0. Q(m) = T (T + 2) m k=1 ˆρ 2 k T k χ 2 (m) 16 / 32
1 1.1 1.2 1.3 1.4 2 ARMA 2.1 ARMA 17 / 32
ARMA ARMA 18 / 32
1 1.1 1.2 1.3 1.4 2 ARMA 2.1 ARMA 19 / 32
- y t y t 1 { y t = a + b y t 1 = b + c - y t y t 1 y t = ay t 1 + b (MA) (AR) 20 / 32
2.1.1 MA MA MA(1) y t = µ + ϵ t + θ 1 ϵ t 1, ϵ t W.N.(σ 2 ) y t MA(1) - y t 1 = µ + ϵ t 1 + θ 1 ϵ t 1 ϵ t 1 θ 1 21 / 32
mu=0, theta1=0.8, sig=1 mu=2, theta1=0.5, sig=1 2 1 0 1 5 0 5 0 20 40 60 80 100 0 20 40 60 80 100 mu=2, theta1=0.3, sig=2 mu=0, theta1=0.3, sig=1 10 5 0 5 0 20 40 60 80 100 2 1 0 1 2 0 20 40 60 80 100 mu=2, theta1=0.5, sig=0.5 mu=2, theta1=0.8, sig=2 10 5 0 5 6 2 0 2 4 6 0 20 40 60 80 100 0 20 40 60 80 100 22 / 32
MA(1) µ E(y t ) = E(µ + ϵ t + θ 1 ϵ t 1 ) = µ + 0 + θ 1 0 = µ. ϵ γ 0 = V ar(y t ) = V ar(µ + ϵ t + θ 1 ϵ t 1 ) = V ar(ϵ t + θ 1 ϵ t 1 ) = V ar(ϵ t ) + θ1v 2 ar(ϵ t 1 ) + 2θ 1 Cov(ϵ t, ϵ t 1 ) = σ 2 + θ1σ 2 2 + 0 = (1 + θ1)σ 2 2. θ 2 1σ 2 ϵ 23 / 32
MA(1) γ 1 = Cov(y t, y t 1 ) = Cov(µ + ϵ t + θ 1 ϵ t 1, µ + ϵ t 1 + θ 1 ϵ t 2 ) = Cov(ϵ t + θ 1 ϵ t 1, ϵ t 1 + θ 1 ϵ t 2 ) = Cov(ϵ t, ϵ t 1 ) + Cov(ϵ t, θ 1 ϵ t 2 ) + Cov(θ 1 ϵ t 1, ϵ t 1 ) + Cov(θ 1 ϵ t 1, θ 1 ϵ t 2 ) ( 0) = θ 1 Cov(ϵ t 1, ϵ t 1 ) = θ 1 σ 2. ρ 1 = γ1 γ 0 = θ1 1+θ 2 1 0 t MA(1) 24 / 32
MA(q) q y t = µ + ϵ t + θ 1 ϵ t 1 + + θ q ϵ t q, ϵ t W.N(σ 2 ). 1 E(y t ) = µ. 2 γ 0 = V ar(y t ) = (1 + θ1 2 + + θ2 q)σ 2. 3 { (θ k + θ 1 θ k+1 + + θ q k θ q )σ 2, 1 k q γ k = 0, k q + 1 4 MA 5 ρ k = { θk +θ 1 θ k+1 + +θ q k θ q 1+θ 2 1 + +θ2 q, 1 k q 0, k q + 1 25 / 32
AR MA(q) AR AR AR(1) y t = c + ϕ 1 y t 1 + ϵ t, ϵ t W.N(σ 2 ) 26 / 32
y y 0 20 40 60 80 100 y y y y c=2,phi=0.8,sigma=1 c= 2,phi=0.3,sigma = 0.5 c=0,phi= 0.3,sigma=2 0 5 10 15 4 3 2 1 0 6 4 2 0 2 4 6 Index 0 20 40 60 80 100 Index 0 20 40 60 80 100 Index c= 2,phi= 0.8,sigma=1 c=2,phi=1,sigma=1 c=2,phi=1.1,sigma=1 6 4 2 0 2 4 0 50 100 150 200 0 5000 10000 15000 20000 0 20 40 60 80 100 Index 0 20 40 60 80 100 Index 0 20 40 60 80 100 Index 27 / 32
MA AR AR(1) ϕ 1 < 1 MA AR c µ = E(y t ) = E(c + ϕ 1 y t 1 + ϵ t ) = c + ϕ 1 E(y t 1 ) = c + ϕ 1 µ µ = c 1 ϕ 1 MA σ 2 γ 0 = V ar(y t ) = V ar(c + ϕ 1 y t 1 + ϵ t ) = V ar(ϕ 1 y t 1 + ϵ t ) = ϕ 2 1V ar(y t 1 ) + V ar(ϵ t ) + 2Cov(y t 1, ϵ t ) = ϕ 2 1 = ϕ 2 1γ 0 + σ 2 γ 0 = σ 2 /(1 ϕ 2 1). 28 / 32
AR(1) k γ k = Cov(y t, y t k ) = Cov(ϕ 1 y t 1 + ϵ t, y t k ) = Cov(ϕ 1 y t 1, y t k ) + Cov(ϵ t, y t k ) = ϕ 1 γ k 1 γ 0 ρ k = ϕ 1 ρ k 1 ( ) - AR ρ 0 = 1 (AR(1) ρ k = ϕ k 1) 29 / 32
AR(p) p AR AR(p) y t = c + ϕ 1 y t 1 + + ϕ p y t p + ϵ t, ϵ t W.N.(σ 2 ). AR(p) 1 µ = E(y t ) = c 1 ϕ 1 ϕ 2 ϕ p. 2 γ 0 = V ar(y t ) = σ 2 1 ϕ 1 ρ 1 ϕ pρ p. 3 y t AR p γ k = ϕ 1 γ k 1 + + ϕ p γ k p, k 1 ρ k = ϕ 1 ρ k 1 + + ϕ p ρ k p, k 1 4 AR 30 / 32
2.1.3 ARMA (ARMA) AR MA ARMA(p,q) y t = c + ϕ 1 y t 1 + + ϕ p y t p + ϵ t + θ 1 ϵ t 1 + θ p ϵ t p, ϵ t W.N(σ 2 ). - AR MA ARMA Ex. MA AR ARMA 31 / 32
ARMA ARMA(p,q) 1 µ = E(y t ) = c 1 ϕ 1 ϕ 2 ϕ p. 2 q+1 p γ k = ϕ 1 γ k 1 + + ϕ p γ k p, k q + 1 ρ k = ϕ 1 ρ k 1 + + ϕ p ρ k p, k 1 + 1 q MA 3 ARMA 32 / 32