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AUTOREGRESSIVE MODELS IN BIG DATA PROBLEMS
Denis Shchepakin, Kegan Rabil,Peter Golubtsov
University of Montana2015
ABSTRACT LINEAR MODEL WITH MEMORY
ut+1 = A
ut−τ+1ut−τ+2ut
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
+ ε t
ui ∈k
A :k ×k ××k → k
ε i iid(0,S)
ut+1 = Awt + ε t
A : k × kτ
ut+1 = Awt + ε t
ut1+1 ut2+1 utγ +1⎡⎣⎢
⎤⎦⎥= A wt1
wt2 wtγ
⎡⎣⎢
⎤⎦⎥+ ε t1 ε t2 ε tγ⎡⎣⎢
⎤⎦⎥
U = AW +Ε
A : k × kτ
U : k × γ W : kτ × γ
E : k × γ
ABSTRACT LINEAR MODEL WITH MEMORY
U = AW +Ε
A :minE AW −U2
A =UW †
Ε = ε t1 ε t2 ε tγ⎡⎣⎢
⎤⎦⎥
ε i iid(0,S)
ABSTRACT LINEAR MODEL WITH MEMORY
S = 1γ −τ
ε tnε tnT
n=1
γ
∑= 1γ −τ
U − AW( ) U − AW( )T
E
BIG DATA ADJUSTMENTS
W : kτ × γ
A =UW †
Problems?
1. is huge.
2. Want to update.
γ
S = 1γ −τ
U − AW( ) U − AW( )T
BIG DATA ADJUSTMENTSA =UW † =UW T WW T( )−1
P =UW T : k × kτR =WWT : kτ × kτ
A = PR−1
Q =UUT : k × k
S = 1γ −τ
UUT −UW T WW T( )−1WUT( )
S = 1γ −τ
Q − PR−1PT( )
BIG DATA ADJUSTMENTSU1 = AW1 +V1 U1 : k × γ 1 W1 : kτ × γ 1 V1 : k × γ 1
U2 : k × γ 2 W2 : kτ × γ 2 V2 : k × γ 2
andP1 R1 Q1 P2 R2 Q2 → P R Q
U1 = AW1 + E1
U2 = AW2 + E2
E1 : k × γ 1
E2 : k × γ 2
U1 U2⎡⎣
⎤⎦ = A W1 W2
⎡⎣
⎤⎦ + E1 E2⎡
⎣⎤⎦
BIG DATA ADJUSTMENTS
P =UW T = [ U1 U2 ]W1
T
W2T
⎡
⎣⎢⎢
⎤
⎦⎥⎥
=U1W1T +U2W2
T = P1 + P2
R =WWT = W1 W2⎡⎣
⎤⎦
W1T
W2T
⎡
⎣⎢⎢
⎤
⎦⎥⎥=W1W1
T +W2W2T = R1 + R2
Q =UUT = U1 U2⎡⎣
⎤⎦
U1T
U2T
⎡
⎣⎢⎢
⎤
⎦⎥⎥=U1U1
T +U2U2T =Q1 +Q2
LONG TERM PREDICTION
A = A1 A2 Aτ⎡⎣
⎤⎦
ut+1 = A
ut−τ+1ut−τ+2ut
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
+ ε t
ui ∈k
A :k ×k ××k → k
ε i iid(0,S)
A : k × kτ
ut+1 = A1ut−τ+1 + A2ut−τ+2 ++ Aτut + ε t
LONG TERM PREDICTIONut+1 = A1ut−τ+1 ++ Aτ−1ut−1 + Aτutut+2 = A1ut−τ+2 ++ Aτ−1ut + Aτ ut+1ut+3 = A1ut−τ+3 ++ Aτ−1ut+1 + Aτ ut+2
νn = ut+n − ut+n
ν1 = ε tν2 = ε t+1 + Aτε tν3 = ε t+2 + Aτ Aτε t + ε t+1( )+ Aτ−1ε t
Fn : kτ × kτF0F1 = S
- zero matrix
LONG TERM PREDICTION
Fn = BFn−1BT +
S 0 00 0 0 0 0 0
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
B =
A1 A2 Aτ−1 Aτ
I 0 0 00 I 0 0 0 0 I 0
⎡
⎣
⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥
Fn : kτ × kτF0F1 = S
- zero matrix