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Transcript of 2013.06.17 Time Series Analysis Workshop ..Applications in Physiology, Climate Change and Finance,...
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Linear stochastic processes Linear time series (stochastic process)
t i t i
i
X Z
2~ WN(0, )t ZZ | |i
i
? E[ ] 0tX
2( )X i i
i
We assume μ=0
A linear time series is expressed as …
Considering the lag operator :
( )t tX B Z ( ) i
i
i
B B
If for i<0 0i
1 1 2 2
1
t t t t t i t i
i
X Z Z Z Z Z
1 moving average process MA(∞)
tXtZ( )B
Linear filter
1 1 2 2
1
t t t t i t i t
i
X X X Z X Z
2 autoregressive process AR(∞)
is stationary tX
0
| |i
i
is invertible tX the stochastic component
can be expressed in terms of
the current and past
observations
tZ
tX
1( )
( )t t t tB X Z X Z
B
1( )
( )B
B
Time series, Part 2
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Autoregressive processes
Autoregression process of order p, AR(p)
2~ WN(0, )t ZZ 1 1 2 2t t t tX X X Z
We restrict the autoregression to the first p most recent terms
1 1 2 2t t t p t p tX X X X Z
Condition of stationarity
The roots of must be outside the unit circle
or
the roots of must be inside the unit circle 1
1 1 0p p
p p
( ) 0
2
1 2(1 )p
p t tB B B X Z ( ) t tB X Z
1
( ) 1p
i
i
i
B B
2
1 2( ) 1 p
pB B B B characteristic polynomial
1 1 2 2
1
t t t t i t i t
i
X X X Z X Z
2 autoregression AR(∞)
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Autoregressive process of order one, AR(1)
2
1 2
0
i
t t t t t i
i
X
Successive backward substitutions: 2
2 2 4 2 2 2
02
Var[ ] (1 )1
i
t
i
XX
2
1 1 11 1 1 1E[ ] E[ ] E[ )] 1) 1( (t t t t t tt t Xt t t X XX X Z X X X X X ZX X
Autocorrelation? (we assume stationarity)
1 1E[ ] E[ ] E[ ] ( ) ( 1)t t t t t t t t tt X XtX X Z X X XX X ZX X
( )X
1t t tX X Z 2~ WN(0, )t ZZ Stationarity condition: | | 1
0 2 4 6 8 10-1
-0.5
0
0.5
1
(
)
()
0 2 4 6 8 10-1
-0.5
0
0.5
1
(
)
()
0.8 0.8
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Autoregressive process of order two, AR(2)
1 1 2 2t t t tX X X Z 2~ WN(0, )t ZZ
2
1 1 2
1,2
2
4
2
Ρίζες:
2 1
1,2 2 1
2
1
1 1
1 1
?
-3 -2 -1 0 1 2 3-3
-2
-1
0
1
2
3
1
2
Stationarity condition for AR(2)
real distinct roots
complex roots
real single root
two real roots: 2
1 24 0
complex conjugate roots: 2
1 24 0
one double real root: 2
1 24 0
The roots of must be outside the unit circle
Stationarity condition
2
1 2( ) 1B B B
The roots of must be inside the unit circle 2
1 2 0
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0 5 10 15 20-1
-0.5
0
0.5
1
(
)
()
1=1.6
2=-0.89
0 5 10 15 20-1
-0.5
0
0.5
1
(
)
()
1=-1.6
2=-0.89
(α) λ1=0.8+0.5i
λ2=0.8-0.5i
(β) λ1=-0.8+0.5i
λ2=-0.8-0.5i
0 5 10 15 20-1
-0.5
0
0.5
1
(
)
()
1=1.6
2=-0.64
0 5 10 15 20-1
-0.5
0
0.5
1
(
)
()
1=-1.6
2=-0.64
(γ) λ1=0.8
λ2=0.8
(δ) λ1=-0.8
λ2=-0.8
0 5 10 15 20-1
-0.5
0
0.5
1
(
)
()
1=1.75
2=-0.76
0 5 10 15 20-1
-0.5
0
0.5
1
(
)
()
1=-0.15
2=0.76
0 5 10 15 20-1
-0.5
0
0.5
1
(
)
()
1=0.15
2=0.76
0 5 10 15 20-1
-0.5
0
0.5
1
(
)
()
1=-1.75
2=-0.76
(ε) λ1=0.8
λ2=0.95
(στ) λ1=0.8
λ2=-0.95
(ζ) λ1=-0.8
λ2=0.95
(η) λ1=-0.8
λ2=-0.95
Autocorrelation
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Autoregressive process of order two, AR(2)
1 2 21 1 1t tt tt tX XX X ZX
Autocorrelation ? (we assume stationarity)
1 1 1 1 2 1 2 1E[ ] E[ ] E[ ] E[ ]t t t t t t t tX X X X X X X Z
1
2
21 2(1) (1) (1) (1)X XX X X
1 1 2 1
2 2 22 1 1t tt tt tX XX X ZX
2 1 2 1 2 2 1 2E[ ] E[ ] E[ ] E[ ]t t t t t t t tX X X X X X X Z
2
2 1 21(2) (1 (2) (1))X X X X X
2 1 1 2
11
2
2
12 2
2
1
1
1 21 2
1
2
2 12 2
1
(1 )
1
1
Για υστέρηση τ:
1 2( ) ( 1) ( 2)X X X
1 1 2 2
can be
computed
recursively
2
1 2(1 ) 0B B
characteristic
polynomial
real roots: exponential decay
complex roots: decaying harmonic
function
1 1 2 2t tt tt tX ZXX X X 2 2
1 2(1) (2)X X X Z
22
1 1 2 21
ZX
variance
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Autoregressive process of order p, AR(p)
2~ WN(0, )t ZZ 1 1 2 2t t t p t p tX X X X Z
2
1 2(1 )p
p t tB B B X Z
Roots of must be outside the unit circle
Stationarity condition
2
1 2( ) 1 p
pB B B B
Autocorrelation ? (we assume stationarity)
For lag τ:
1 1 2 2t t t p t p tt tX X X X ZX X
1 2( ) ( 1) ( 2) ( )X X X p X p
1 1 2 2 2E[ ] E[ ] E[ ] E[ ] E[ ]t t t t t t p t p t t p tX X X X X X X X X Z
1 2( ) ( 1) ( 2) ( ) ( ) 0X X X p X p B
real roots : exponential decay
complex roots : decaying harmonic function
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Autoregressive process of order p, AR(p)
1 1 2 2 p p
1
2
p
1 1 2 1 1
2 1 1 2 2
1 1 2 2
p p
p p
p p p p
Yule-Walker
equations
1
2
p
1
2
p
p
1 2 1
1 1 2
1 2 3
1
1
1
p
p
p
p p p
p p
1
p p
1 1 2 2t t t p tt t p tX X XX X ZX
2 2
1 2(1) (2) ( )X X X p X Zp 2
2
1 1 2 21
ZX
p p
Variance
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Partial autocorrelation
Yule-Walker
equations
1 2 1 1 1
1 1 2 2 2
1 2 3
1
1
1
k
k
k k k k k
For each k we compute
the coefficient k kk
1k 111
2k
1
21 2 2 1
1
2 2
1
1
2
1
1 1
1
3k
1 1
1 2
2 1 3
1
3
1 1
2 1
3
2
1
1
1
1
1
12 11 2
1231 1
11 2 31
21 2 1
131 1 2
111 2 3
k
k
kkk
k k k
k k
k k
k k k
partial autocorrelation for lag (order) k
Recursive algorithm of Durbin-Levinson
the coefficients of AR(p) 1 2, , , ,p p pp
are computed recursively, and for each order k
the coefficients are computed from the
coefficients of order k-1
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(α) λ1=0.8+0.5i
λ2=0.8-0.5i
(β) λ1=-0.8+0.5i
λ2=-0.8-0.5i
(γ) λ1=0.8
λ2=0.8
(δ) λ1=-0.8
λ2=-0.8
(ε) λ1=0.8
λ2=0.95
(στ) λ1=0.8
λ2=-0.95
(ζ) λ1=-0.8
λ2=0.95
(η) λ1=-0.8
λ2=-0.95
0 5 10 15 20-1
-0.5
0
0.5
1
(
, )
()
1=1.6
2=-0.89
0 5 10 15 20-1
-0.5
0
0.5
1
(
, )
()
1=-1.6
2=-0.89
0 5 10 15 20-1
-0.5
0
0.5
1
(
, )
()
1=1.6
2=-0.64
0 5 10 15 20-1
-0.5
0
0.5
1
(
, )
()
1=-1.6
2=-0.64
0 5 10 15 20-1
-0.5
0
0.5
1
(
, )
()
1=1.75
2=-0.76
0 5 10 15 20-1
-0.5
0
0.5
1
(
, )
()
1=-0.15
2=0.76
0 5 10 15 20-1
-0.5
0
0.5
1
(
, )
()
1=0.15
2=0.76
0 5 10 15 20-1
-0.5
0
0.5
1
(
, )
()
1=-1.75
2=-0.76
Partial autocorrelation
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Moving average processes
Moving average process of order q, ΜΑ(q)
invertibility condition
The roots of must be outside the unit circle
( ) 0
2
1 2(1 )q
t q tX B B B Z ( )t tX B Z 2
1 2( ) 1 q
qB B B B
characteristic polynomial
1 moving average MA(∞)
2~ WN(0, )t ZZ
We constrain the white noise terms to the first q most recent terms
1 1 2 2t t t tX Z Z Z
1 1 2 2t t t t q t qX Z Z Z Z i i
1 1 2 2
1
t t t t t i t i
i
X Z Z Z Z Z
ΜΑ(q) is stationary ?
ΜΑ(q) is invertible if 1( )t tZ B X
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Moving average process of order one, MA(1)
2~ WN(0, )t ZZ Invertibility condition: | | 1 1t t tX Z Z
1 1
2 2 2(1 )... X Zt tt t t tX Z ZX Z Z
? 2
1 1 2 1 21 (1. . )1
.t t t Xt t t ZX Z ZX Z Z
2 3 12 ...t t tt t tX ZX Z Z Z (2) 0X
21
1
0 2
1| | 1/ 2
For one there are two solutions for θ
and only one satisfies the invertibility condition 1
?
10.4t t tX Z Z
12.5t t tX Z Z
Example 1
12.9
0 2
1t t tX Z Z 11/t t tX Z Z and
they have the same autocorrelation
1 0B If the root of is outside
the unit circle
the root of is inside
the unit circle 1 1/ 0B
![Page 13: 2013.06.17 Time Series Analysis Workshop ..Applications in Physiology, Climate Change and Finance, part 2](https://reader035.fdocuments.us/reader035/viewer/2022081720/558d44f2d8b42a39318b467e/html5/thumbnails/13.jpg)
Moving average process of order one, MA(1)
Partial autocorrelation
11 1 21
2 2
12,2 2 2 4
11 1
3 3
13,3 2 2 4 6
11 2 1
2
, 2( 1)
(1 ), 1
1
0 2 4 6 8 10-1
-0.5
0
0.5
1
(
)
()
0 2 4 6 8 10-1
-0.5
0
0.5
1
(
, )
()
0 2 4 6 8 10-1
-0.5
0
0.5
1
(
)
()
0 2 4 6 8 10-1
-0.5
0
0.5
1
(
, )
()
0.8
0.8
- ϕττ of ΜΑ(1) decays
the same as ρτ of AR(1)
- ρτ of ΜΑ(1) decays
the same as ϕττ of AR(1)
- … but for MA(1),
ρτ and ϕττ are always ≤0.5
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Moving average process of order two, MA(2)
2
1 1 2 2( ) , ~ WN(0, )t t t t t t ZX B Z Z Z Z Z
2
1 2( ) 1B B B
MA(2) is always stationary
MA(2) is invertible if the roots of θ(Β) are outside the unit circle
11 1
2
2 12,2 2
11
3
1 1 2 23,3 2 2
2 1 2
(2 )
1 2 (1 )
2 2 2 2
1 2(1 )X Z Variance
1 2
2 2
1 2
2
2 2
1 2
(1 )1
1
21
0 2
Autocorrelation Partial autocorrelation
, ... complicated
expression
characteristic polynomial
![Page 15: 2013.06.17 Time Series Analysis Workshop ..Applications in Physiology, Climate Change and Finance, part 2](https://reader035.fdocuments.us/reader035/viewer/2022081720/558d44f2d8b42a39318b467e/html5/thumbnails/15.jpg)
Autocorrelation
Partial autocorrelation
λ1=0.8+0.5i
λ2=0.8-0.5i
0 5 10 15 20-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
(
)
()
1=1.6
2=-0.89
0 5 10 15 20-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
(
, )
()
1=1.6
2=-0.89
λ1=-0.8+0.5i
λ2=-0.8-0.5i
0 5 10 15 20-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
(
)
()
1=-1.6
2=-0.89
0 5 10 15 20-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
(
, )
()
1=-1.6
2=-0.89
λ1=0.8
λ2=0.95
0 5 10 15 20-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
(
)
()
1=1.75
2=-0.76
0 5 10 15 20-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
(
, )
()
1=1.75
2=-0.76
λ1=0.8
λ2=-0.95
0 5 10 15 20-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
(
)
()
1=-0.15
2=0.76
0 5 10 15 20-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
(
, )
()
1=-0.15
2=0.76
- ϕττ of ΜΑ(2) decays
same as ρτ of AR(2) - ρτ of ΜΑ(2) decays
same as ϕττ of AR(2)
- … but for MA(2),
ρτ and ϕττ is always ≤0.5
![Page 16: 2013.06.17 Time Series Analysis Workshop ..Applications in Physiology, Climate Change and Finance, part 2](https://reader035.fdocuments.us/reader035/viewer/2022081720/558d44f2d8b42a39318b467e/html5/thumbnails/16.jpg)
Moving average process of order q, MA(q)
2~ WN(0, )t ZZ 1 1 2 2( )t t t t t q t qX B Z Z Z Z Z
2 2 2 2
1(1 )X q Z Variance
Autocovariance
2
1 1( ) 1,2, ,
0
Z q q q
q
Autocorrelation
1 1
2 2 2
1 2
1,2, ,1
0
q q
q
q
q
The partial autocorrelation decays in a way that is determined
from the roots of the characteristic polynomial
2
1 2( ) 1 q
qB B B B characteristic polynomial
The expressions of ϕττ in terms of the
coefficients θ1, θ2, ..., θq are complicated
![Page 17: 2013.06.17 Time Series Analysis Workshop ..Applications in Physiology, Climate Change and Finance, part 2](https://reader035.fdocuments.us/reader035/viewer/2022081720/558d44f2d8b42a39318b467e/html5/thumbnails/17.jpg)
2~ WN(0, )t ZZ 1 1 2 2t t t p t p tX X X X Z
2
1 2(1 )p
p t tB B B X Z
( ) t tB X Z
Autoregressive process
order p, AR(p)
1 1 2 2t t t t q t qX Z Z Z Z
Moving average process
of order q, ΜΑ(q)
( )t tX B Z
2
1 2(1 )q
t q tX B B B Z 2
1 2( ) 1 q
qB B B B 2
1 2( ) 1 p
pB B B B
ΜΑ(∞) AR(∞) ( )t tX B Z
1
( )t tX B Z
2
1 2( ) 1B B B
such that ( ) ( ) 1B B
( ) t tB X Z
1
( ) t tB X Z
2
1 2( ) 1B B B
such that ( ) ( ) 1B B
AR(p) ↔ MA(∞) MA(q) ↔ AR(∞) AR(p) and MA(q)
have dual relation
The autocorrelation and partial autocorrelation of
AR(p) and MA(q) have also dual relation
AR(p): ρτ decays exponentially to 0, ϕττ gets zero for τ>p
MA(q): ϕττ decays exponentially to 0, ρτ gets zero for τ>q
Wold's decomposition (1)
every covariance-stationary
time series can be written
as an infinite moving
average (MA(∞)) process
of its innovation process.
AR(p) stationary MA(q) invertible
Relation between AR and MA processes
![Page 18: 2013.06.17 Time Series Analysis Workshop ..Applications in Physiology, Climate Change and Finance, part 2](https://reader035.fdocuments.us/reader035/viewer/2022081720/558d44f2d8b42a39318b467e/html5/thumbnails/18.jpg)
Autoregressive moving average process ARMA(p,q)
2~ WN(0, )t ZZ
Autoregressive process
of order p, AR(p)
1 1 2 2t t t t q t qX Z Z Z Z
Moving average process
of order q, ΜΑ(q)
1 1 2 2t t t p t p tX X X X Z
1 1 2 2 1 1 2 2t t t p t p t t t q t qX X X X Z Z Z Z
( ) ( )t tB X B Z
1 1 2 2 1 1 2 2t t t p t p t t t q t qX X X X Z Z Z Z
( )
( )t t
BX Z
B
( )
( )t t
BX Z
B
Stationarity is determined by the AR part
Invertibility is determined by the MA part
Autocorrelation:
1 1 2 2 1 1 2 2( )t t t p t p t t t q tt qtX X X X ZX X Z Z Z
1 1 1( ) ( 1) ( ) E[ ] E[ ] E[ ]X X p X t t t t q t t qp X X X
For q 1 1 p p
1 1 p p such as for AR(p)
Για q mixing of autocorrelation for AR(p) and MA(q)
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Process ARMA(1,1)
2
1
( )(1 )1
1 2
2
?
1 1t t t tX X Z Z (1 ) (1 )t tB X B Z (1 )
(1 )t t
BX Z
B
Invertibility condition: | | 1
Stationarity condition: | | 1
1 1( )t t t tt tX XX X Z Z
1 1E[ ] E[ ]t t t tX X
2 2 2
0 1 ( )X Z Z 0
2
1 0 Z 1
22
0 2
1 2
1Z
2
1 2
( )(1 )
1Z
1 1 such as for AR(1)
Autocorrelation:
Partial autocorrelation : decays with the lag such as for MA(1)
An ARMA(p,q) process with small p,q, exhibits correlation pattern (ρτ and ϕττ)
that can be attained by AR(p) only for large order p, or
by MA(q) only for large order q
![Page 20: 2013.06.17 Time Series Analysis Workshop ..Applications in Physiology, Climate Change and Finance, part 2](https://reader035.fdocuments.us/reader035/viewer/2022081720/558d44f2d8b42a39318b467e/html5/thumbnails/20.jpg)
Estimation of models AR, MA, ARMA
(stationary) time series
(stochastic process) t tX
(stationary) time series
of n observations 1 2, , , nx x x
autocovariance
2
(( ) )( )t t
t t t
X X
X X X
sample autocovariance
2
1
(1
( ))n
t t
t
x x xn
c c
0,1, , 1n
1
1 n
t
t
xxn
sample mean value mean value μ
(
(0)(
))
cr r
c
sample autocorrelation autocorrelation
( )
0)(
()
stochastic process AR(p)
1 1 2 2t t t p t p tX X X X Z 2~ WN(0, )t ZZ
stochastic process MA(q)
1 1 2 2t t t t q t qX Z Z Z Z
1 1 2 2
1 1 2 2
t t t p t p t
t t q t q
X X X X Z
Z Z Z
stochastic process ARMA(p,q)
Estimation of the process (model)
● order p or/and q ?
● estimation of model parameters ?
? 2
1 2AR( ) : , , , ,pp
2
1 2ΜΑ( ) : , , , ,qq
2
1 2 1 2ARΜΑ( , ) : , , , , , , , ,p qp q
● AR, MA or ARMA ? other model ?
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Estimation of model AR(p)
1 2, , , nx x xWe assume a stochastic process AR(p) generate the time series
Fit of a model AR(p) estimation of parameters 2
1 2, , , ,p
Method of moments or method of Yule-Walker (YW)
Estimation of the parameters from the sample autocorrelations 2
1 2, , , ,p Xr r r s 2
1 2ˆ ˆ ˆ, , , ,p s
2
1 2, , , ,p Xr r r sEstimation of 2
1 2, , , ,p X and then substitution …
Yule-Walker
equations
1 2 1 1 1
1 1 2 2 2
1 2 3
1
1
1
p
p
p p p p p
22
1 1 2 21
ZX
p p
p p
11 2 1 1
1 1 2 22
1 2 3
ˆ1
ˆ1
1 ˆ
p
p
p p p pp
r r r r
r r r r
r r r r
22
1 1 2 2ˆ ˆ ˆ1
ZX
p p
ss
r r r
ˆp pR r
1ˆp p
R r 2 2
1 1 2 2ˆ ˆ ˆ(1 )Z X p ps s r r r
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The estimation method of ordinary least squares (OLS)
1 2, , , nx x x with a mean μ
General form of AR(p) 1 1 2 2( ) ( ) ( )t t t p t p tX X X X Z
Fit of model AR(p) to the data
Minimization of the sum of squares of the fitting errors
2
1 1 1
1
min ( , , ) min ( ) ( )n
p t t p t p
t p
S x x x
w.r.t. 1 2, , , , p
1 2ˆ ˆ ˆˆ , , , , p
1 1ˆ ˆˆ ˆ ˆˆ ( ) ( ) ( ), 1, ,t t t p t pz x x x t p n
2
1
2 1ˆ
n
t
t p
Z zsn p
1
ˆ1 n
t
t
xn
x
AR(1) 1 1( )t t tX X Z
2
1 1 1
2
( , ) ( )n
t t
t
S x x
(2) (1)ˆ
ˆˆ1
x x
(2)
2
1
1
n
t
t
x xn
(1) 1
2
1
1
n
t
t
x xn
1 12 2
2 2
2 2
ˆ ˆ( )( ) ( )( )ˆ
ˆ( ) ( )
n n
t t t tt t
n n
t tt t
x x x x x x
x x x
ˆ x
1 12
1( )( )
n
t ttc x x x x
n
2
0 1
1( )
n
ttc x x
n
11
0
ˆc
rc
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Other methods for estimation of AR(p)
● backward – forward approach (FB)
● maximum likelihood (ML)
- conditioned
- unconditioned
● Burg’s algorithm (Burg)
The ML estimation is optimal, the other methods approximate it
The ML reduces to OLS when the time series is from a Gaussian process
Asymptotically (for large n) all methods converge to the same (ML) estimates
The YW has the slowest convergence rate to ML
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Determination of order p of an AR model
1,1 1t t tx x z
1,2 1 2,2 2t t t tx x x z
1,3 1 2,3 2 3,3 3t t t t tx x x x z
estimation of for model AR(τ)
partial autocorrelation for lag τ: 1 1, , , ,t tt tx xx x accounting for the correlation with
correlation of
The order is p if and for τ>p ,ˆ 0p p ,
ˆ 0 (fall from non-zero to zero
partial autocorrelation)
the criterion of partial autocorrelation
the criterion based on fitting errors
● 2 2AIC( ) ln( )z
pp s
n Akaike information criterion (AIC)
● 2 ln( )BIC( ) ln( )z
p np s
n Bayesian information criterion (BIC)
● 2FPE( ) z
n pp s
n p
Final prediction error (FPE)
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Παράδειγμα Growth rate of gross national product (GNP) of USA
quarter-annual observations, 2nd quarter 1947 – 1st quarter 1991).
The time series is corrected for seasonality
0 50 100 150-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
t
xt
GNP of USA: increments
0 5 10 15 20
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
( )
incr.GNO(USA): autocorrelation
0 5 10 15 20
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
,
incr.GNO(USA): partial autocorrelation
0 5 10 15 20
-9.26
-9.24
-9.22
-9.2
-9.18
-9.16
p
AIC
(p)
incr.GNO(USA): AIC
stationary ?
correlation ?
AR(3) ?
order of
AR model ?
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parameter estimation
ˆ 0.0077 ˆt tx x
OLS
0 1 2 3ˆ ˆ ˆ ˆˆ 1 0.0047
1ˆ 0.35 2
ˆ 0.18 3ˆ 0.14
1 2 30.0047 0.35 0.18 0.14t t t t tx x x x z fitted AR(3)
1 2 3ˆ 0.0047 0.35 0.18 0.14t t t tx x x x 4, ,176t estimation
ˆ 0.0098z zs ˆ
t̂ t tz x x errors or residual of fit 2 2ˆ 0.0000989z zs
Diagnostic check for model adequacy
Is the residual time series independent? test for independence on 1
ˆn
t t pz
0 50 100 150 200-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
time t
x(t)
incr.GNP(USA): AR(3) fit
100 110 120 130 140-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
time tx(t
)
incr.GNP(USA): AR(3) fit
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1 2, , , nx x xWe assume a stochastic process MA(q) for the time series
Fit of the process (model) MA(q) parameter estimation 2
1 2, , , ,q
Fit of the model MA(q)
stochastic process AR(p)
1 1 2 2t t t p t p tX X X X Z 2~ WN(0, )t ZZ
stochastic process MA(q)
1 1 2 2t t t t q t qX Z Z Z Z
1 1 2 2
1 1 2 2
t t t p t p t
t t q t q
X X X X Z
Z Z Z
stochastic process ARMA(p,q)
Estimation of the process (model)
● order p or/and q ?
● estimation of model parameters ?
? 2
1 2AR( ) : , , , ,pp
2
1 2ΜΑ( ) : , , , ,qq
2
1 2 1 2ARΜΑ( , ) : , , , , , , , ,p qp q
● AR, MA or ARMA ? other model ?
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Method of moments
Autocorrelation
1 1
2 2 2
1 2
1,2, ,1
0
q q
q
q
q
2 2 2 2
1(1 )X q Z Variance
Nonlinear equation system
w.r.t. the parameters 1 2, , , q
2
1 2, , , ,q Xr r r sEstimation of 2
1 2, , , ,q X
Method of ordinary least squares
Fit of model MA(q) to the data
Minimization of sum of squares of fitting errors
2
1 1 1
1
min ( , , ) min ( )n
q t t q t q
t q
S x z z
w.r.t. 1 2, , , , q
1 2ˆ ˆ ˆ, , , q
1 1 2 2t t t t q t qX Z Z Z Z MA(q)
Numerical optimization method
Innovation algorithm
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MA(1) 1t t tX Z Z
Method of moments
2
1
1
1
1 1 ,2
2
1ˆ 1 1 4ˆ| | 0 5 ˆ
2. 0
r
rr r r
11
1
ˆ| | 0.5| |
rr
r
We choose the solution that gives rise to invertibility ˆ| | 1
21
1
0 2
2 2 2(1 )q
X Z 2
2
2ˆ1
XZ
ss
Method of ordinary least squares
1 1z x
2 2 1 2 1z x z x x
3 3 2 3 2 1 3 2 1
2( )z x z x x x x x x
2 2 2 2 2 1 2
1 2 1 3 2 1 1 1
1
min min ( ) ( ) ( )n
n
t n n
t
z x x x x x x x x x
2 2
0 1 2 2min n
na a a
2 2
1 1
1
2 2 1n n n n n
n
n
nz x z x x x x x
computational algorithm: least
squares with constraints for
invertibility
2n-2 solutions, we select the
solution ˆ| | 1 0 0z 0 We assume (and ) 1t t tz x z
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Παράδειγμα Growth rate of gross national product (GNP) of USA
quarter-annual observations, 2nd quarter 1947 – 1st quarter 1991).
The time series is corrected for seasonality
0 50 100 150-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
t
xt
GNP of USA: increments
0 5 10 15 20
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
r()
incr.GNP(USA): autocorrelation
ΜΑ(2) ?
order of the
MA model ?
0 2 4 6 8 10-9.24
-9.22
-9.2
-9.18
-9.16
-9.14
q
AIC
(q)
incr.GNP(USA): AIC of MA models
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parameter estimation
1 20.0077 0.312 0.272t t t tx z z z 1, ,176t fitted ΜΑ(2)
0.00983zs variance of errors (residuals) 2 0.000097zs
0.0077x
OLS 1ˆ 0.312 2
ˆ 0.272
0 50 100 150 200-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
time t
x(t)
incr.GNP(USA): MA(2) fit
100 110 120 130 140-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
time t
x(t
)
incr.GNP(USA): MA(2) fit
fit with
ΜΑ(2)
0 50 100 150 200-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
time t
x(t)
incr.GNP(USA): AR(3) fit
100 110 120 130 140-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
time t
x(t
)
incr.GNP(USA): AR(3) fit
fit with
AR(3)
Diagnostic check for model adequacy
Is the residual time series independent? test for independence on 1
ˆn
t t pz
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1 2, , , nx x xWe assume a stochastic process ARMA(p,q) for the time series
Fit of the process (model) ARMA(p,q)
estimation of parameters 2
1 2 1 2, , , , , , , ,p q
Εκτίμηση μοντέλου ARMA(p,q)
stochastic process AR(p)
1 1 2 2t t t p t p tX X X X Z 2~ WN(0, )t ZZ
stochastic process MA(q)
1 1 2 2t t t t q t qX Z Z Z Z
1 1 2 2
1 1 2 2
t t t p t p t
t t q t q
X X X X Z
Z Z Z
stochastic process ARMA(p,q)
Estimation of the process (model)
● order p or/and q ?
● estimation of model parameters ?
? 2
1 2AR( ) : , , , ,pp
2
1 2ΜΑ( ) : , , , ,qq
2
1 2 1 2ARΜΑ( , ) : , , , , , , , ,p qp q
● AR, MA or ARMA ? other model?
The methods of moments and least squares as for MA(q)
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Method of ordinary least squares
1 1z x
2 2 1 1 2 1( )z x x z x x
We assume (and ) 0 0z 0 0x
3 3 2 2 3 2 1( ) ( )z x x z x x x
2
1 1 1 2 1( ) ( ) ( )n
n n n n n n nz x x z x x x x
ARMA(1,1) 1 1( )t t t tX X Z Z
Solution of equation system w.r.t. ,
2
1 2, , , ,p Xr r r sEstimation of 2
1 2, , , X Method of moments
2
1
( )(1 )1
1 2
2
22 2
2
1 2
1X Z
22 2
2
ˆ1
ˆ ˆ ˆ1 2Z Xs s
?
2
1
minn
t
t
z
computational algorithm of least squares with
constraints for invertibility and stationarity
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Παράδειγμα Growth rate of gross national product (GNP) of USA
quarter-annual observations, 2nd quarter 1947 – 1st quarter 1991).
The time series is corrected for seasonality
0 50 100 150-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
t
xt
GNP of USA: increments
0 5 10 15 20
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
r()
incr.GNP(USA): autocorrelation
0 2 4 6 8 10
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
p
p,p
incr.GNP(USA): partial autocorrelation
-1 0 1 2 3 4 5 6-9.24
-9.22
-9.2
-9.18
-9.16
-9.14
p
AIC
(p,q
)
incr.GNP(USA): AIC of ARMA models
q=0
q=1
q=2
q=3
q=4
q=5
ARMA(2,2) ?
order of
ARMA ?
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0 50 100 150 200-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
time t
x(t)
incr.GNP(USA): AR(3) fit
100 110 120 130 140-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
time t
x(t
)
incr.GNP(USA): AR(3) fit
0 50 100 150 200-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
time t
x(t)
incr.GNP(USA): MA(2) fit
100 110 120 130 140-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
time t
x(t
)
incr.GNP(USA): MA(2) fit
fit with
ΜΑ(2)
fit with
AR(3)
parameter estimation
OLS
1 2 1 2ˆ 0.0065 0.614 0.455 0.301 0.600t t t t t tx x x z z z 1, ,176t fitted ARΜΑ(2,2)
0.00983zs variance of errors (residuals) 2 0.000097zs
0.0077x
1ˆ 0.614 1
ˆ 0.301 2ˆ 0.600 2
ˆ 0.455
0 50 100 150 200-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
time t
x(t)
incr.GNP(USA): ARMA(2,2) fit
100 110 120 130 140-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
time t
x(t
)
incr.GNP(USA): ARMA(2,2) fit
fit with
ARΜΑ(2,2)
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Model for time series with trends (ARIMA)
E 0tX 2 2E tX
random walk
1 21t t t tX X XY Y X
1t t
Y
t tX
iid
process AR(1) for 1
(non-stationary process)
First differences: 1(1 )t t t tX B Y Y Y iid process
1t t
Y
non-stationary process that exhibits trends
first differences: 1t t tX Y Y stationary process? NO
second order differences: 1 1 22t t t t t tX X X Y Y Y
stationary process ?
YES
YES
AR(p), MA(q), ARMA(p,q) ?
NO
1t t
Y
non-stationary process ARIMA(p,d,q)
1 1 2 2 1 1 2 2t t t p t p t t t q t qX X X X Z Z Z Z
( ) ( )t tB X B Z
( ) ( )d
t tB Y B Z
stationary after d order differences: 1t t
X
d
t tX Y
(1 )d
tB Y
The polynomial has a
unit root and all other roots are
outside the unit circle
( )(1 )dB B
( )(1 ) ( )d
t tB B Y B Z
Usually 1d
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Fit of model ARIMA (Box-Jenkins approach)
1 2, , , ny y ytime series
indication that there is trend
? autocorrelation (strong and slowly decaying)
other ?
?
1 2, , , nx x xstationary time series
d-order differences
other ? (1 )d
t tx B y
fit of model AR(p), MA(q), ARMA(p,q)
model order
estimation of model parameters
model adequacy
diagnostic test
model ARMA(p,q) for 1 2, , , nx x x
then using the inverse transform (1 )d
t tx B y
we get the model ARΙMA(p, d,q) for 1 2, , , ny y y
time series history diagram
if autocorrelation
decays to zero
the time series
is stationary
if the autocorrelation
is statistically not
significant
is it iid ?
STOP
test for
independence YES
NO
nonlinear
model ?
?
prediction ?
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0 5 10 15-0.2
0
0.2
0.4
0.6
0.8
r()
annual global temperature: autocorrelation
1840 1860 1880 1900 1920 1940 1960 1980 2000 2020-1
-0.5
0
0.5
1
year
d(t
em
p)
first differences of annual land air temperature anomalies
0 5 10 15-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
r()
first difference of annual global temperature: autocorrelation
1840 1860 1880 1900 1920 1940 1960 1980 2000 2020-1
-0.5
0
0.5
1
1.5
year
glo
bal te
mpera
ture
annual land air temperature anomalies
1 2, , , ny y y real observations
Example Annual index of global temperature (anomaly of surface temperature
of the north hemisphere at grid 5ο x 5ο), time period 1850-2011 Source: http://www.cru.uea.ac.uk/cru/data/temperature
stationary
time series?
stationary
time series?
1 2, , , nx x x first differences
NO
YES
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Model for time series ? 1 2, , , nx x x
0 5 10 15-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
r()
first difference of annual global temperature: autocorrelation
0 5 10 15-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
( )
diff of temp: partial autocorrelation
-1 0 1 2 3 4 5 6-3.25
-3.2
-3.15
-3.1
-3.05
-3
-2.95
-2.9
-2.85
p
AIC
(p,q
)
diff of temp: AIC of ARMA models
q=0
q=1
q=2
q=3
q=4
q=5
autocorrelation partial autocorrelation AIC criterion
The most appropriate model ?
1840 1860 1880 1900 1920 1940 1960 1980 2000 2020-1
-0.5
0
0.5
1
time t
x(t
)
diff of global temperature: ARMA(0,4) fit
1930 1935 1940 1945 1950 1955 1960-1
-0.5
0
0.5
1
time t
x(t
)
diff of global temperature: ARMA(0,4) fit
fit of ΜΑ(4) ( ) 0.008x
1 2 3 40.008 0.758 0.022 0.219 0.275t t t t t tx z z z z z 0.2035zs 2 0.0414zs
Model for time series 1 2, , , ny y y
ARIΜΑ(0,1,4) 4(1 ) ( )t tB Y B Z
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Model of time series with seasonality (ARMAs)
Hypothesis: there are correlations but only between the same components
of each period (the dependence occurs at time steps s) :
1 2 3 1 2 2 2 1 2 2 3 3 1 3 2, , , , , , , , , , , , , , ,s s s s s s s s s nx x x x x x x x x x x x x
k cycles of period s
Given the time series without trend and with seasonality s 1 2, , , nx x x
s – differences (difference of lag s) (1 )s
t s t t t t sX Y B Y Y Y
Removal of seasonality of period s, : /k n s
Estimation of the periodic components si i=1,…,s 1
1 k
i i js
j
s yk
t t tx y s
Symmetric moving
average of order s /2 /2 1 /2 1 /2
1(0.5 0.5 )t t s t s t s t sx y y y y
s s even
( 1)/2
( 1)/2
1 s
t t i
i s
x ys
s odd
1 2, , , ny y yGiven the time series without trend and with seasonality (periodicity)
1 ( 1) ( 1) 1 ( 1) ( 1)i st i s t P i s P i st i s t Q i s QX X X Z Z Z
model ARMA(P,Q)s for 2, , , ,i i s i s i ksx x x x the same for 1,2, ,i s
1,2, ,i s
1 1t t s P t Ps t t s Q t QsX X X Z Z Z 1, 2, ,t Ps Ps n
( ) ( )s s
t tB X B Z model ARMA(P,Q)s for 1 2, , , nx x x
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Given the time series with seasonal trend and
given that the correlations are between components with the same periodic order
Model of time series with seasonality (ARIMAs)
1 2, , , ny y y
This is an extension of ARMA(P,Q)s when the time series has “seasonal trend”,
meaning trend at the time points t, t+s, t+s, …
s – differences (difference of lag s)
(1 )s
t s t t t t sx y B y y y
1 2, , ,s s nx x x 1 2, , , ny y y
ARMA(P,Q)s ( ) ( )s s
t tB X B Z
ARIMA(P,1,Q)s
( )(1 ) ( )s s s
t tB B Y B Z
In general, ARIMA(P,D,Q)s
( )(1 ) ( )s s D s
t tB B Y B Z
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Example Mean monthly temperature at Thessaloniki station, period 1930-2000
YR JAN FEB MAR APR MAY JUN ΙJUL AUG SEP OCT NOV DEC
1930 6,7 6,7 11,3 15,7 19 22,6 26,2 26 22,8 17,5 12,1 8,9
1931 7,9 8,8 10 12,7 19,7 24,9 27,4 26,9 21,5 16 10,8 4,1
1932 5 2,9 7,4 14,1 19,4 24,1 27,2 26,1 24,1 21,5 11,7 9,2
1933 5,2 7,6 9,1 13,5 17,6 22,8 25,5 25,3 20,5 17,7 14,2 5,5
1934 5,3 5,7 12,6 15,9 20,5 23,9 26,9 26,3 23,1 17,6 13,4 10
1935 4,5 6,8 8,2 14,7 18,7 24,9 26,1 26 22,8 19,8 12,1 9,2
1936 10,5 8,3 12,2 16 18,4 23 27,1 25,9 21,6 16,3 12,3 7,2
1937 4,8 8,2 12,9 14,5 19,7 24,4 26,4 26 23,6 16,9 12,8 8
1938 4,8 6,6 11 12,9 18,4 24,1 27,5 27,1 22,2 17,9 12,8 8,8
1939 7,9 7,9 8,5 15,5 19,8 23,1 27,2 26,6 21,9 18,5 12,3 7,7
1940 3,1 6,8 8,1 13,9 17,5 22,9 26,6 24,3 21,4 18,5 13 4,2
1941 6,9 10,2 11 15,6 19,2 23,7 26 26 18,9 15,7 10,1 4,8
1942 0,9 5,6 8,6 13,9 20,7 24,6 25,4 26,1 23,8 17,5 10,7 7,3
part of the record
30 35 40 45 50 55 60 65 70 75 80 85 90 95 00 050
5
10
15
20
25
30
year
Tem
p
Temperature Thessaloniki, period 1/1930-12/2000
1930 1940 1950 1960 1970 1980 1990 20000
5
10
15
20
25
30
year
Tem
p
Temperature Thessaloniki, period 1/1930-12/2000 - month
1 2, , , nx x x 71*12 852n
removal of
seasonality
30 35 40 45 50 55 60 65 70 75 80 85 90 95 00 05-6
-4
-2
0
2
4
6
year
Tem
p
Temp Thess: subtract average period
Estimation of seasonal component
30 35 40 45 50 55 60 65 70 75 80 85 90 95 00 0514.5
15
15.5
16
16.5
17
17.5
18
year
Tem
p
Temp Thess: moving average with order 12
moving average
30 35 40 45 50 55 60 65 70 75 80 85 90 95 00 05-10
-5
0
5
10
year
Tem
p
Temp Thess: 12-differencing
12-order difference
same model
ARMA(P,Q)s
for each month
?
0 20 40 60 80 100-1
-0.5
0
0.5
1
r()
Temperature Thessaloniki: autocorrelation
strong seasonality
(periodicity)
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600
5
10
15
20
25
30
time t
x(t
)
Temp Thess: ARMA(1,1)12
fit
600
5
10
15
20
25
30
time t
x(t
)
Temp Thess: ARMA(1,1)12
fit
-1 0 1 2 3 4 5 62
2.5
3
3.5
4
4.5
5
p
AIC
(p,q
)
Temp Thess: AIC of ARMA12
models
q=0
q=1
q=2
q=3
q=4
q=5
model ARMA(P,Q)s
30 35 40 45 50 55 60 65 70 75 80 85 90 95 00 050
5
10
15
20
25
30
time t
x(t
)
Temp Thess: ARMA(1,1)12
fit
Fit of ARMA(1,1)12
15.928x
12 120.0075 0.9995 0.5242t t t tx x z z
1.932zs 2 3.733zs
1.603zs
standard deviation
of residual time
series
Fit with the estimation of
the periodic component 1.427zs
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Model of time series with trend and seasonality (SARIMA)
removal of trend removal of seasonality
( ) ( )(1 ) (1 ) ( ) ( )s d s D s
t tB B B B Y B B Z
1 2, , , nx x x 1 2, , , ny y y
dependence between successive
observations (time step 1)
2 1 1 2, , , ,t t t t tx x x x x
dependence between seasonal
components of the same seasonal
order (time step s)
2 2, , , ,t s t s t t s t sx x x x x
SARIMA(p,d,q)×(P,D,Q)s
Seasonal multiplicative model
1 2, , , ny y yGiven that the time series has
trend and seasonality s
ARIMA(p,1,q)
( )(1 ) ( )d
t tB B Y B Z ( )(1 ) ( )s s D s
t tB B Y B Z
ARIMA(P,1,Q)s
most often
1d
0D
SARMA(p,q)×(P, Q)s 0d 0D and
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0 20 40 60 80 100-0.2
0
0.2
0.4
0.6
0.8
1
1.2
r()
monthly global temperature: autocorrelation
50 60 70 80 90 00 10 20 30 40 50 60 70 80 90 00 10 20-3
-2
-1
0
1
2
3
year
glo
bal te
mpera
ture
land air temperature anomalies, period 1/1850-12/2011
1 2, , , ny y y real observations
Example Monthly index of global temperature (anomaly of surface temperature
of the north hemisphere at grid 5ο x 5ο), time period 1850-2011 Source: http://www.cru.uea.ac.uk/cru/data/temperature
1840 1860 1880 1900 1920 1940 1960 1980 2000 2020-3
-2
-1
0
1
2
3
year
glo
bal te
mpera
ture
land air temperature anomalies, period 1/1850-12/2011
Jan
May
Sep
removal of trend ?
removal of seasonality / periodicity ?
dependences between successive
observations (time step 1) ?
dependence between seasonal
components of the same seasonal
order (time step s) ?
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Jan50 Jan52 Jan54 Jan56 Jan58 Jan60 Jan62-1.5
-1
-0.5
0
0.5
1
year
d(t
em
p)
first differences of monthly global temperature
first differences
Jan50 Jan52 Jan54 Jan56 Jan58 Jan60 Jan62-1.5
-1
-0.5
0
0.5
1
1.5
year
d(t
em
p)
first differences of month global temperature
differences of lag 12
significant autocorrelations
for τ=1,2,…
for τ=12,24,…
0 20 40 60 80 100-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
r()
first difference of monthly global temperature: autocorrelation
0 20 40 60 80 100-0.6
-0.4
-0.2
0
0.2
0.4
r()
12-difference of monthly global temperature: autocorrelation
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0 20 40 60 80 100-0.4
-0.3
-0.2
-0.1
0
0.1
( )
diff of temp: partial autocorrelation
-1 0 1 2 3 4 5 6
-1.6
-1.55
-1.5
-1.45
-1.4
-1.35
-1.3
-1.25
p
AIC
(p,q
)
diff of monthly temp: AIC of SARMA(p,q)x(0,0)
q=0
q=1
q=2
q=3
q=4
-1 0 1 2 3 4 5 6
-1.6
-1.55
-1.5
-1.45
-1.4
-1.35
-1.3
-1.25
p
AIC
(p,q
)
diff of monthly temp: AIC of SARMA(p,q)x(1,0)
q=0
q=1
q=2
q=3
q=4
-1 0 1 2 3 4 5 6
-1.6
-1.55
-1.5
-1.45
-1.4
-1.35
-1.3
-1.25
p
AIC
(p,q
)
diff of monthly temp: AIC of SARMA(p,q)x(1,1)
q=0
q=1
q=2
q=3
q=4
-1 0 1 2 3 4 5 6
-1.6
-1.55
-1.5
-1.45
-1.4
-1.35
-1.3
-1.25
p
AIC
(p,q
)
diff of monthly temp: AIC of SARMA(p,q)x(0,1)
q=0
q=1
q=2
q=3
q=4
-1 0 1 2 3 4 5 6
-1.6
-1.55
-1.5
-1.45
-1.4
-1.35
-1.3
-1.25
p
AIC
(p,q
)
diff of monthly temp: AIC of SARMA(p,q)x(0,2)
q=0
q=1
q=2
q=3
q=4
-1 0 1 2 3 4 5 6
-1.6
-1.55
-1.5
-1.45
-1.4
-1.35
-1.3
-1.25
p
AIC
(p,q
)
diff of monthly temp: AIC of SARMA(p,q)x(2,0)
q=0
q=1
q=2
q=3
q=4
-1 0 1 2 3 4 5 6
-1.6
-1.55
-1.5
-1.45
-1.4
-1.35
-1.3
-1.25
p
AIC
(p,q
)
diff of monthly temp: AIC of SARMA(p,q)x(2,3)
q=0
q=1
q=2
q=3
q=4
-1 0 1 2 3 4 5 6
-1.6
-1.55
-1.5
-1.45
-1.4
-1.35
-1.3
-1.25
p
AIC
(p,q
)
diff of monthly temp: AIC of SARMA(p,q)x(1,2)
q=0
q=1
q=2
q=3
q=4
min(AIC)=-1.622 for SARMA(3,3) (1,2)12 SARMA(1,2) (1,1)12 AIC=-1.618
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50 60 70 80 90 00 10 20 30 40 50 60 70 80 90 00 10 20-4
-2
0
2
4
time t
x(t
)
diff global temp: ARMA(3,3)x(1,2)12
fit
60-4
-2
0
2
4
time t
x(t
)
diff global temp: ARMA(3,3)x(1,2)12
fit
SARMA(3,3) (1,2)12
1 2 3 12 13 14 15
1 2 3 12 13 14 15
24 25 26 27
1.12 0.70 0.22 0.95 1.11 0.70 0.18
0.42 0.22 0.95 1.01 0.48 0.23 0.93
0.13 0.08 0.05 0.08
t t t t t t t t
t t t t t t t t
t t t t
x x x x x x x x
z z z z z z z z
z z z z
0.445zs
0.0013x
50 60 70 80 90 00 10 20 30 40 50 60 70 80 90 00 10 20-4
-2
0
2
4
time t
x(t
)
diff global temp: ARMA(1,2)x(1,1)12
fit
60-4
-2
0
2
4
time t
x(t
)
diff global temp: ARMA(1,2)x(1,1)12
fit
SARMA(1,2) (1,1)12
1 12 13
1 2 12 13 14
0.35 0.98 0.34
1.04 0.1 0.93 0.99 0.12
t t t t
t t t t t t
x x x x
z z z z z z
0.446zs
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Prediction of time series
Index and volume of the Athens Stock Exchange (ASE)
Models for time series (AR, MA, ARMA, ARIMA, SARIMA) prediction
Many applications
![Page 50: 2013.06.17 Time Series Analysis Workshop ..Applications in Physiology, Climate Change and Finance, part 2](https://reader035.fdocuments.us/reader035/viewer/2022081720/558d44f2d8b42a39318b467e/html5/thumbnails/50.jpg)
Can we predict the index or volume the first day(s) of May 2002
given the observations until the end of April 2002?
![Page 51: 2013.06.17 Time Series Analysis Workshop ..Applications in Physiology, Climate Change and Finance, part 2](https://reader035.fdocuments.us/reader035/viewer/2022081720/558d44f2d8b42a39318b467e/html5/thumbnails/51.jpg)
At what level GICP is to be moved in the next months?
General index of consumer prices (GICP)
01 02 03 04 05 06100
105
110
115
120
125
years
Genera
l In
dex o
f C
om
sum
er
Prices
General Index of Comsumer Prices, period Jan 2001 - Aug 2005
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Given the number of sunspots up
to the current date, how many
sunspots will be next year(s)?
Sunspots
1700 1750 1800 1850 1900 1950 20000
50
100
150
200
years
num
ber
of
sunspots
Annual sunspots, period 1700 - 2001
1900 1920 1940 1960 1980 2000
20
40
60
80
100
120
140
160
180
200
years
num
ber
of
sunspots
Annual sunspots, period 1900 - 2001 1960 1970 1980 1990 20000
50
100
150
200
years
num
ber
of
sunspots
Annual sunspots, period 1960 - 2001
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Heart rate
What is the next heart rate(s) ?
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The problem of time series prediction • We are given the time series up to time n
• We want to estimate xn+k
Prediction xn(k)
Prediction error:
nxxx ,,, 21
)()( kxxke nknn
Stochastic process
prediction Xn(k) is the estimation of the observation Xn+k of
}{ nX
}{ nX
1( ) E[ | , , ]n n k n nX k X X X Best prediction :
Properties of a good prediction: • unbiasedness :
• efficiency, meaning small prediction error E[ ( )]n n kX k X
Var[ ( )] Var[ ( )]n n k nk X X k
Optimizing both unbiasedness and efficiency
minimization of the mean square prediction error
2
E ( )n k nX X k
![Page 55: 2013.06.17 Time Series Analysis Workshop ..Applications in Physiology, Climate Change and Finance, part 2](https://reader035.fdocuments.us/reader035/viewer/2022081720/558d44f2d8b42a39318b467e/html5/thumbnails/55.jpg)
2
21 1rmse( ) ( ) ( )
1 1
n l k n l k
j j k j
j n j n
k e k x x kl k l k
root mean square error (rmse)
Evaluation of a prediction model :
given also
1 2{ , , , }n n n lx x x
nxxx ,,, 21 Given
prediction model
1( ), ( ), , ( )n n n l kx k x k x k
prediction errors k
time step ahead 1( ), ( ), , ( )n n n l ke k e k e k
( ) ( )j j k je k x x k
, 1, ,j n n n l k
2
21 1mse( ) ( ) ( )
1 1
n l k n l k
j j k j
j n j n
k e k x x kl k l k
Estimation of mean square error (mse)
2
2
1( )
1nrmse( )
1
1
n l k
j k j
j n
n l k
j k
j n
x x kl k
k
x xl k
normalized root mean
square error (nrmse)
nrmse 0
very good prediction
nrmse ≈ 1
prediction at the level
of mean value prediction
learning /
training set test / validation set
Statistical measures of error
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Predictions:
1 / 2( ) Var ( )n nx k c e kprediction limits
Given , we want to evaluate the predictability
of a prediction model lnnn xxxxx ,,,,,, 121
Given , we predict )(,),2(),1( kxxx nnn nxxx ,,, 21
Prediction many steps ahead for a given current time
Prediction at a given time step ahead for different current times
2~ N(0, )t zz 1 / 2 1 / 2c z
2
21 1rmse( ) ( ) ( )
1 1
n l k n l k
j j k j
j n j n
k e k x x kl k l k
3. We compute a statistic of prediction errors
)(,),(),( 1 kxkxkx klnnn 2. We pursue predictions for some time step ahead k
nxxx ,,, 21 1. We estimate the model parameters based on the time series
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ttt zx
Simple prediction techniques
Deterministic trend (revisited)
trend, a small varying function of time
white noise ),0WN(~ 2
ztz
1 1( ) E | , , ,n n k n k n n n kx k z x x x Prediction:
knknkn zx
Solution: Extrapolation of function μt for times > n
knn zke )(Prediction error:
μt = ?
known simple substitution
unknown estimation
m
mm tctcctp 10)(e.g. polynomial
global (fit to ) nxxx ,,, 21
local (fit only to the m last observations)
nmnmn xxx ,,, 21
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Index and volume of ASE, prediction with trend extrapolation (polynomial fit of trend)
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01 02 03 04 05 06100
105
110
115
120
125
years
Genera
l In
dex o
f C
om
sum
er
Prices
General Index of Comsumer Prices, period Jan 2001 - Aug 2005
56
1{ }t tx
Deterministic seasonal term ttt zsx
deterministic seasonal term and deterministic trend tttt zsx
Same approach: estimation of the deterministic term
t t t tx s z
01 02 03 04 05 06-4
-3
-2
-1
0
1
2
3
years
detr
ended G
ICP
General Index of Comsumer Prices, linear trend is subtracted
t t tx x
01 02 03 04 05 06-4
-3
-2
-1
0
1
2
3
4
years
year
cycle
of
GIC
P
General Index of Comsumer Prices, year cycle
1
n
t ts
01 02 03 04 05 06-4
-3
-2
-1
0
1
2
3
4
years
detr
ended a
nd d
eseasoned G
ICP
General Index of Comsumer Prices, trend and period comp. subtracted
t t t t t tz x s x s
( )n n k n kx k s
103.9 + 0.31t t
57 103.9 + 0.31*57 121.70
Prediction of Sept 2005
9 0.16s
56(1) 121.86x
GICP, January 2001 – August 2005
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1
01
n
j jc
1
0
11110)(n
j
jnjnnnn xcxcxcxckx
Exponential smoothing
Estimation of xn+k as a weighted sum of former observations
110 nccc Desired condition on the weights:
10,1,,1,0,)1( njc j
j
Determination of the weights
with a single parameter :
)()1()( 1 kxxkx nnn
recursive relation :
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Index and volume of ASE
Prediction at one time step ahead for all days in May 2002
Comparison of the prediction performance of exponential smoothing for different
Large (weighting most the most recent observations) gives the best prediction
Prediction with exponential smoothing: Examples
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Heart rate
Sunspots
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Predictions with AR, MA and ARMA
Prediction with autoregressive models (AR)
AR(1) model ttt zxx 1
n
k
n xkx )(
white noise ),0WN(~ 2
ztz
11 nnn zxx 1 nt
Optimal prediction at time step 1: nn xx )1(
Optimal prediction at time step 2:
212 nnn zxx 2 nt
nnn xxx 2)1()2(
Optimal prediction at time step k:
nxxx ,,, 21 Given the time series
knt
1)1( nn ze
Prediction error:
2Var (1)n ze
Prediction error :
21)2( nnn zze
2 2Var (2) ( 1)n ze
2
2
2
1Var ( )
1
k
n ze k
Prediction error :
knknn
k
n zzzke
11
1)(
Prediction of stationary time series with linear models
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Optimal prediction at time step 2:
2121 )1()2( pnpnnn xxxx
nxxx ,,, 21 Given the time series
22112 npnpnn zxxx 2 nt
1)1( nn ze
Prediction error :
2)1(Var zne
Prediction error :
21121 )1()2( nnnnn zzzee
22 )1()2(Var zne
AR(p) model tptptt zxxx 11
1 nt 1111 npnpnn zxxx
Optimal prediction at time step 1:
11)1( pnpnn xxx
knt
knkpnpknkn zxxx 11
Optimal prediction at time step k:
)()1()( 1 pkxkxkx npnn
prediction ( ) 0where ( )
observation 0
n
n
n j
x j jx j
x j
Prediction error :
knnknn zekeke )1()1()( 1
1
0
)(k
j
jknjn zbke
1
0
22)(Vark
j
jzn bke
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AR(1) AR(6) AR(11)
0.9995 1.1535 1.1523
-0.2126 -0.2131
0.0944 0.0961
-0.0655 -0.0663
0.0103 0.0135
0.0194 -0.0031
-0.0058
0.0190
-0.0175
0.0458
-0.0213
Index ASE, multi-step prediction for May 2002 20,,1,2002.04.30),( knkxn
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Index ASE, one step ahead prediction in May 2002 2002.05.312002.05.2),1( nxn
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Volume ASE, multi-step prediction for May 2002 20,,1,2002.04.30),( knkxn
AR(1) AR(6) AR(11)
0.9097 0.3412 0.3251
0.2092 0.1955
0.1557 0.1380
0.1369 0.1138
0.0773 0.0455
0.0528 0.0009
0.0350
0.0068
0.0249
0.0420
0.0527
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Volume ASE, one step ahead prediction in May 2002 2002.05.312002.05.2),1( nxn
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AR(1) AR(6) AR(11)
0.8205 1.3231 1.1848
-0.5297 -0.4385
-0.1655 -0.1718
0.1895 0.1933
-0.2576 -0.1324
0.1702 0.0311
0.0157
-0.0203
0.1993
-0.0186
0.0352
Sunspots, multi step prediction from 1991 to 2001 21,,1,1990),( knkxn
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Sunspots, prediction one year ahead in period 1991-2001 20011991),1( nxn
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Heart rate, prediction of the next 21 heart rates 21,,1,1060),( knkxn
AR(1) AR(6) AR(11)
0.8065 0.7850 0.7803
-0.1205 -0.0736
0.1983 0.1759
0.1438 0.0858
-0.1407 -0.1239
-0.0465 -0.1899
0.1413
0.0761
0.0073
-0.0463
0.0347
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Heart rate, prediction of the next heart rates 10811061),1( nxn
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Growth rate of GNP of USA
The observations are at annual-quarters, from the second
quarter of 1947 till the first quarter of 1991 (n=176)
0 50 100 150-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
t
x t
Rate growth of GNP of USA
0 5 10 15 20
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
r( )
Autocorrelation of rate growth
0 2 4 6 8 10
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
p
p,p
Partial Autocorrelation for Rate Growth
0 2 4 6 8 10-9.23
-9.22
-9.21
-9.2
-9.19
-9.18
-9.17
-9.16
-9.15
p
AIC
(p)
AIC for Rate Growth
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164 166 168 170 172 174 176-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04Prediction of rate growth with AR(3)
164 166 168 170 172 174 176-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04Prediction of rate growth with AR(1)
( ), 170, 1, ,6nx k n k
1 2 30.0047 0.35 0.18 0.14t t t t tx x x x z
ˆ 0.0077 1ˆ 0.35 2
ˆ 0.18 3ˆ 0.14
0 1 2 3ˆ ˆ ˆ ˆˆ 1 0.0047
ˆ 0.0098z zs
0 1 1 2 2 3 3t t t t tx x x x z AR(3)
AR(1)
10.0047 0.38t t tx x z
ˆ 0.0099z zs
Growth rate of GNP of USA
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AR(p), p=1,…,10
(1), 126 176nx n
predictability for k step ahead
0 2 4 6 8 100.7
0.8
0.9
1
1.1
p
nrm
se(p
)
nrmse(k) on the last 50 data
k=1k=2
(1), 146 176nx n
0 2 4 6 8 100.7
0.8
0.9
1
1.1
p
nrm
se(p
)
nrmse(k) on the last 30 data
k=1k=2
Growth rate of GNP of USA
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Optimal prediction at time step 1: nn zx )1( 1)1( nn ze
Prediction error :
2)1(Var zne
Prediction error :
2)2( nn xe
2
2Var)2(Var xnn xe
MA(1) model 1 ttt zzx
),0WN(~ 2
ztz
0 αν
0 αν0,,|E 1 jz
jxxz
jn
nnjn
1 nt nnn zzx 11
2 nt 122 nnn zzx
Optimal prediction at time step 2: 0)2( nx
1για0
1για)(
k
kzkx
n
n
1για
1για)(
1
kx
kzke
kn
n
n
For time step k:
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1)1( nn ze
Prediction error :
2)1(Var zne
Prediction error :
112)2( nnn zze
22 )1()2(Var zne
Prediction error :
1111)( nkknknn zzzke
1
0
)(k
j
jknjn zke
1
0
22)(Vark
j
jzn ke
MA(q) model qtqttt zzzx 11
1 nt 1111 qnqnnn zzzx
Optimal prediction at time step 1:
11)1( qnqnn zzx
2 nt 221122 qnqnnnn zzzzx
Optimal prediction at time step 2:
22)2( qnqnn zzx
knt qknqknknkn zzzx 11
Optimal prediction at time step k:
1 1 if( )
0 if
k n k n q n q k
n
z z z k qx k
k q
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Growth rate of GNP of USA
The observations are at annual-quarters, from the second
quarter of 1947 till the first quarter of 1991 (n=176)
( ), 170, 1, ,6nx k n k
164 166 168 170 172 174 176-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04Prediction of rate growth with MA(2)
(1), 146 176nx n
0 2 4 6 8 100.7
0.8
0.9
1
1.1
q
nrm
se(q
)
nrmse(k) with MA(q) on the last 30 data
k=1k=2
ΜΑ(2)
1 20.0077 0.41 0.40t t t tx z z z
ˆ 0.0109z zs
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ARMA(p,q) model qtqttptptt zzzxxx 1111
111111 qnqnnpnpnn zzzxxx
1 nt
1111)1( qnqnpnpnn zzxxx
Optimal prediction at time step 1: 1)1( nn ze
Prediction error :
2)1(Var zne
1
1
( 1) ( ) if( )
( 1) ( ) if
n p n k n q n q k
n
n p n
x k x k p z z k qx k
x k x k p k q
Optimal prediction at time step k:
Prediction with ARMA: merging of the prediction with AR and MA
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Growth rate of GNP of USA
1 2 3 1 20.0034 0.15 0.29 0.12 0.33 0.13t t t t t t tx x x x z z z
ARMA(3,2)
ˆ 0.0105z zs
( ), 170, 1, ,6nx k n k
164 166 168 170 172 174 176-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04Prediction of rate growth with ARMA(3,2)
(1), 146 176nx n
0 2 4 6 8 100.7
0.8
0.9
1
1.1
p
nrm
se(p
)
nrmse(k) with ARMA(p,1) on the last 30 data
k=1k=2
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1 2, , , ny y yGiven a non-stationary time series
t t t ty s x standard decomposition model for yt :
n k n k n k n ky s x knt prediction of yn+k :
Estimation of μt and st
as functions of time t Removal of μt and st (using
differences) prediction with
models ARIMA or SARIMA t t t tx y s 1.
2. : prediction (of type ARMA) of xn+k ( )nx k
( ) ( )n n k n k ny k s x k 3.
Stages of prediction:
( )ny k 1 2, , , ny y y1. transformation to stationary
1 2, , , nx x x
1 2. prediction of xn+k with some model
( )nx k2
3. inverse transform on the prediction
3
Prediction of non-stationary time series
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ARIMA(p,1,q)
(1)nx2. prediction of xn+1 with ARMA(p,q)
(1) (1)n n ny y x
(1) (1)n ny x3. inverse transform :
1t t tx y y
1 2 2 3, , , , ,n ny y y x x x
Stages of prediction of :
stationary 1. transformation :
(1)ny
)1(~)1( nn ee
prediction error :
prediction error of (1)nx
For prediction at k steps ahead :
( ) ( 1) ( )n n ny k y k x k
known from the prediction of yn+k-1 ARMA(p,q) prediction of xn+k
Similar procedure for the prediction with models
ARIMA(p,d,q) or SARIMA(p,d,q)(P,D,Q)s
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ASE index Period from January 2002 to September 2005
02 03 04 05 061000
1500
2000
2500
3000
3500
years
clo
se index
ASE General Index, Jan 2002 - Sep 2005
0 10 20 30 40 500
0.2
0.4
0.6
0.8
1
r( )
Autocorrelation of ASE General Index
Returns 1
1
t tt
t
y yx
y
02 03 04 05 06-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
years
clo
se index r
etu
rns
Returns of ASE General Index
0 5 10 15 20-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
r( )
Autocorrelation of returns of ASE General Index
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ASE index Period from January 2002 to September 2005
0 5 10 15 20-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
p
p
,p
Partial autocorrelation of returns of general index
0 5 10 15 20-9.135
-9.13
-9.125
-9.12
-9.115
-9.11
-9.105
p
AIC
(p)
AIC of returns of general index
Prediction of many steps, all for current time on 20/9/2005
18 25 02 09 16-0.015
-0.01
-0.005
0
0.005
0.01
0.015
days
retu
rns o
f in
dex
yn(k) of index return, n=20.9.2005
return of indexy
n(T), AR(7)
returns of ASE
18 25 02 09 163200
3250
3300
3350
3400
3450
days
clo
se index
xn(k) of general index, n=20.9.2005
general indexx
n(T), AR(7)
ASE index
1(1 )t t ty y x
1 1(1 )n n ny y x
Order of AR model 1
1
t tt
t
y yx
y
( ) ( 1)(1 ( ))n n ny k y k x k
(1) (1 (1))n n ny y x
Prediction
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ASE index Period from January 2002 to September 2005
18 25 02 09 163200
3250
3300
3350
3400
3450
days
clo
se index
xn(1) of general index n=20.9.2005 to 12.10.2005
general indexAR(1)AR(7)
One step ahead prediction for period
20/9/2005 – 12/10/2005
0 5 10 15 200.5
1
1.5
p
nrm
se(p
)
nrmse of AR for general index, 20.9.2005-12.10.2005
k=1k=2k=5
Estimation of prediction error
with ΑR(p) models
for the period 20/9/2005 – 12/10/2005