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 ADJUSTMENTS

A, S A, S

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

BIG DATA ADJUSTMENTS

A = PR−1

P1, R1, Q1, γ 1

P2, R2, Q2, γ 2

P, R, Q, γ

S = 1γ −τ

Q − PR−1PT( )

APPLICATION

Days

Celcius ×10 Min temperature

APPLICATION

Days

Max temperatureCelcius ×10

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

APPLICATION

Days

Celcius ×10 Min temperature

APPLICATION

Days

Max temperatureCelcius ×10

Thank you!