time series analysis. definition a time series {x t : t t} is a collection of random variables...
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![Page 1: Time Series Analysis. Definition A Time Series {x t : t T} is a collection of random variables usually parameterized by 1) the real line T = R = (-∞,](https://reader035.vdocuments.us/reader035/viewer/2022062518/56649e525503460f94b480f7/html5/thumbnails/1.jpg)
Time Series Analysis
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Definition
A Time Series {xt : t T} is a collection of random variables usually parameterized by
1) the real line T = R= (-∞, ∞)2) the non-negative real line T = R+ = [0, ∞)3) the integers T = Z = {…,-2, -1, 0, 1, 2, …}
4) the non-negative integers T = Z+ = {0, 1, 2, …}
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If xt is a vector, the collection of random vectors
{xt : t T}
is a multivariate time series or multi-channel time series.
If t is a vector, the collection of random variables
{xt : t T} is a multidimensional “time” series or spatial series.
(with T = Rk= k-dimensional Euclidean space or a k-dimensional lattice.)
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Example of spatial time series
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The project
• Buoys are located in a grid across the Pacific ocean
• Measuring– Surface temperature– Wind speed (two components)– Other measurements
The data is being collected almost continuously
The purpose is to study El Nino
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Technical Note:The probability measure of a time series is defined by specifying the joint distribution (in a consistent manner) of all finite subsets of {xt : t T}.
i.e. marginal distributions of subsets of random variables computed from the joint density of a complete set of variables should agree with the distribution assigned to the subset of variables.
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The time series is Normal if all finite subsets of
{xt : t T} have a multivariate normal distribution.
Similar statements are true for multi-channel time series and multidimensional time series.
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Definition:(t) = mean value function of {xt : t T} = E[xt]
for t T.
(t,s) = covariance function of {xt : t T}
= E[(xt - (t))(xs - (s))] for t,s T.
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For multichannel time series(t) = mean vector function of {xt : t T} = E[xt] for t T and(t,s) = covariance matrix function of {xt : t T}
= E[(xt - (t))(xs - (s))′] for t,s T.
The ith element of the k × 1 vector (t) i(t) =E[xit]
is the mean value function of the time series {xit : t T}
The i,jth element of the k × k matrix (t,s) ij(t,s) =E[(xit - i(t))(xjs - j(s))]
is called the cross-covariance function of the two time series {xit : t T} and {xjt : t T}
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Definition:
The time series {xt : t T} is stationary if the joint distribution of xt1
, xt2, ... , xtk
is the
same as the joint distribution of xt1+h ,xt2+h , ... ,xtk+h for all finite subsets t1,
t2, ... , tk of T and all choices of h.
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Definition:
The multi-channel time series {xt : t T} is stationary if the joint distribution of xt1
,
xt2, ... , xtk
is the same as the joint distribution
of xt1+h , xt2+h , ... , xtk+h for all finite subsets
t1, t2, ... , tk of T and all choices of h.
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Definition:
The multidimensional time series {xt : t T} is stationary if the joint distribution of xt1
, xt2,
... , xtk is the same as the joint distribution of
xt1+h ,xt2+h , ... ,xtk+h for all finite subsets t1,
t2, ... , tk of T and all choices of h.
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Time
The distribution of observations at these points in time
The distribution of observations at these points in timesame as
Stationarity
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Some Implication of Stationarity
If {xt : t T} is stationary then:
1. The distribution of xt is the same for all t T.
2. The joint distribution of xt, xt + h is the same as the joint distribution of xs, xs + h .
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Implication of Stationarity for the mean value function and the covariance function
If {xt : t T} is stationary then for t T.
(t) = E[xt] = and for t,s T.
(t,s) = E[(xt - )(xs - )]
= E[(xt+h - )(xs+h - )]
= E[(xt-s - )(x0 - )] with h = -s= (t-s)
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If the multi-channel time series{xt : t T} is stationary then for t T.
(t) = E[xt] =
and for t,s T
(t,s) = (t-s)
Thus for stationary time series the mean value function is constant and the covariance function is only a function of the distance in time (t – s)
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If the multidimensional time series {xt : t T} is stationary then for t T.
(t) = E[xt] =
and for t,s T.
(t,s) = E[(xt - )(xs - )]
= (t-s) (called the Covariogram)
Variogram
V(t,s) = V(t - s) = Var[(xt - xs)] = E[(xt - xs)2]
= Var[xt] + Var[xs] –2Cov[xt,xs]
= 2[(0) - (t-s)]
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Definition:(t,s) = autocorrelation function of {xt : t T}
= correlation between xt and xs.
for t,s T.
sstt
st
xx
xx
st
st
,,
,
varvar
,cov
If {xt : t T} is stationary then
(h) = autocorrelation function of {xt : t T}
= correlation between xt and xt+h.
o
h
oo
h
xx
xx
tht
tht
varvar
,cov
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Definition:
The time series {xt : t T} is weakly stationary if:
(t) = E[xt] = for all t T.
and
(t,s) = (t-s) for all t,s T.
or
(t,s) = (t-s) for all t,s T.
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Examples
Stationary time series
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1. Let X denote a single random variable with mean and standard deviation . In addition X may also be Normal (this condition is not necessary)
Let xt = X for all t T = { …,, -2, -1, 0, 1, 2, …}
Then E[xt] = = E[X] for t T and
(h) = E[(xt+h - )(xt - )]
= Cov(xt+h,xt )
= E[(X - )(X - )] = Var(X)
= 2 for all h.
. allfor 1 ho
hh
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Excel file illustrating this time series
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2. Suppose {xt : t T} are identically distributed and uncorrelated (independent).T = { …,, -2, -1, 0, 1, 2, …}
Then E[xt] = for t T and
(h) = E[(xt+h - )(xt - )]
= Cov(xt+h,xt )
00
0
h
hxVar t
00
02
h
h
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The auto correlation function:
00
01
h
h
o
hh
Comment:
If = 0 then the time series {xt : t T} is called a white noise time series.
Thus a white noise time series consist of independent identically distributed random variables with mean 0 and common variance 2
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Excel file illustrating this time series
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3. Suppose X1, X2, … , Xk and Y1, Y2, … , Yk are independent independent random variables with
sincos1
k
iiiiit tYtXx
222 and iii YEXE 0 ii YEXE
Let 1, 2, … k denote k values in (0,)
For any t T = { …,, -2, -1, 0, 1, 2, …}
2sin2cos1
k
iiiii tYtX
2
sin2
cos1
k
i ii
ii P
tY
P
tX
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Excel file illustrating this time series
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Then
sincos1
k
iiiiit tYtXExE
tht xxEh
0 sincos1
k
iiiii tYEtXE
sincos1
k
iiiii htYhtXE
sincos1
k
jjjjj tYtX
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Hence
coscos1 1
k
i
k
jjiji thtXXEh
thtYY jiji sinsin
sincos thtYX jiji cossin thtXY jiji
sinsincoscos1
2
k
iiiiii thttht
if 0 0,0 since jiYYEXXEYXE jijiji
and 222iii YEXE
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Hence using cos(A – B) = cos(A) cos(B) + sin(A) sin(B)
k
iii
k
iiii hthth
1
2
1
2 cos cos
and
k
iiik
jj
k
iii
hwh
hh
1
1
2
1
2
coscos
0
k
jj
iiw
1
2
2
where
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4. The Moving Average Time series of order q, MA(q)
qtqtttt uuuux 22110
Let 0 =1, 1, 2, … q denote q + 1 numbers.
Let {ut|t T} denote a white noise time series with variance 2.
– independent– mean 0, variance 2.
Let {xt|t T} be defined by the equation.
qtqttt uuuu 2211
Then {xt|t T} is called a Moving Average time series of order q. MA(q)
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Excel file illustrating this time series
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The mean
qtqtttt uuuuExE 22110
t h th E x x
qtqttt uEuEuEuE 22110
The auto covariance function
qhtqhththt uuuuE 2211
qtqttt uuuu 2211
q
jjtj
q
iihti uuE
00
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q
i
q
jjtihtji uuE
0 0
q
i
q
jjtihtji uuE
0 0
qi
qihq
ihii
0
if0
2
. if 0 since jiuuE ji . and 22 iuE
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qi
qih
hq
ihii
0
if0
2
The autocorrelation function for an MA(q) time series
The autocovariance function for an MA(q) time series
qi
qihh
q
ii
hq
ihii
0
if0 0
2
0
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5. The Autoregressive Time series of order p, AR(p)
Let 1, 2, … p denote p numbers.
Let {ut|t T} denote a white noise time series with variance 2.
– independent– mean 0, variance 2.
Let {xt|t T} be defined by the equation.
2211 tptpttt uxxxx
Then {xt|t T} is called a Autoregressive time series of order p. AR(p)
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Excel file illustrating this time series
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Comment:
where {ut|t T} is a white noise time series with variance 2. i.e. 1 = 1 and = 0.
11 ttt uxx
An Autoregressive time series is not necessarily stationary.
Suppose {xt|t T} is an AR(1) time series satisfying the equation:
1 tt ux
121 tttttt uuxuxx
tt uuuux 1210
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and is not constant.
ttt uEuEuEuExExE 1210
0xE
tt uVaruVarxVarxVar 10
but
20 txVar
A time series {xt|t T} satisfying the equation:
is called a Random Walk.
1 ttt uxx
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Derivation of the mean, autocovariance function and autocorrelation function of a
stationary Autoregressive time series
We use extensively the rules of expectation
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is stationary.
Assume that the autoregressive time series {xt|t T} be defined by the equation:
2211 tptpttt uxxxx
Let = E(xt). Then
2211 tptpttt uExExExExE
21 p
1 21 p
p
txE
211
or
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The Autocovariance function, (h), of a stationary
autoregressive time series {xt|t T}can be determined by using the equation:
2211 tptpttt uxxxx
Thus
1 2Now 1 p
11 tptptt uxxx
The Autocovariance function, (h)
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Hence
tht xxEh
thtphtpht xuxxE 11
where
tht xxE 11
thttphtp xuExxE
hphh uxp 11
0
00
hxuE
hxuEh
ttthtux
![Page 44: Time Series Analysis. Definition A Time Series {x t : t T} is a collection of random variables usually parameterized by 1) the real line T = R = (-∞,](https://reader035.vdocuments.us/reader035/viewer/2022062518/56649e525503460f94b480f7/html5/thumbnails/44.jpg)
Now
0ux t tE u x
1 1t t p t p tE u x x u
211 tpttptt uExuExuE
2
![Page 45: Time Series Analysis. Definition A Time Series {x t : t T} is a collection of random variables usually parameterized by 1) the real line T = R = (-∞,](https://reader035.vdocuments.us/reader035/viewer/2022062518/56649e525503460f94b480f7/html5/thumbnails/45.jpg)
The equations for the autocovariance function of an AR(p) time series
21 10 pp
101 1 pp
212 1 pp
323 1 pp
etc
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Or using (-h) = (h)
21 10 pp
101 1 pp
212 1 pp
and
011 ppp
phhh p 11 for h > p
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Use the first p + 1 equations to find (0), (1) and (p)
Then use
phhh p 11
for h > pTo compute (h)
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The Autoregressive Time series of order p, AR(p)
Let 1, 2, … p denote p numbers.
Let {ut|t T} denote a white noise time series with variance 2.
– independent– mean 0, variance 2.
Let {xt|t T} be defined by the equation.
2211 tptpttt uxxxx
Then {xt|t T} is called a Autoregressive time series of order p. AR(p)
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is stationary.
If the autoregressive time series {xt|t T} be defined by the equation:
2211 tptpttt uxxxx
Then
p
txE
211
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The Autocovariance function, (h), of a stationary
autoregressive time series {xt|t T} be defined by the equation:
2211 tptpttt uxxxx
Satisfy the equations:
![Page 51: Time Series Analysis. Definition A Time Series {x t : t T} is a collection of random variables usually parameterized by 1) the real line T = R = (-∞,](https://reader035.vdocuments.us/reader035/viewer/2022062518/56649e525503460f94b480f7/html5/thumbnails/51.jpg)
21 10 pp
101 1 pp
212 1 pp
and
011 ppp
phhh p 11 for h > p
Yule Walker Equations
The autocovariance function for an AR(p) time series
The mean 1 21t
p
E x
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Use the first p + 1 equations (the Yole-Walker Equations) to find (0), (1) and (p)
Then use
phhh p 11
for h > pTo compute (h)
![Page 53: Time Series Analysis. Definition A Time Series {x t : t T} is a collection of random variables usually parameterized by 1) the real line T = R = (-∞,](https://reader035.vdocuments.us/reader035/viewer/2022062518/56649e525503460f94b480f7/html5/thumbnails/53.jpg)
The Autocorrelation function, (h), of a stationary
autoregressive time series {xt|t T}:
0
hh
The Yule walker Equations become:
![Page 54: Time Series Analysis. Definition A Time Series {x t : t T} is a collection of random variables usually parameterized by 1) the real line T = R = (-∞,](https://reader035.vdocuments.us/reader035/viewer/2022062518/56649e525503460f94b480f7/html5/thumbnails/54.jpg)
011
2
1 pp
111 1 pp
212 1 pp
and
111 ppp
phhh p 11 for h > p
![Page 55: Time Series Analysis. Definition A Time Series {x t : t T} is a collection of random variables usually parameterized by 1) the real line T = R = (-∞,](https://reader035.vdocuments.us/reader035/viewer/2022062518/56649e525503460f94b480f7/html5/thumbnails/55.jpg)
pp
110
1
2
111 1 pp
212 1 pp
Then
111 ppp
phhh p 11
for h > p
To find (h) and (0): solve for (1), …, (p)
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Example
Consider the AR(2) time series:
xt = 0.7xt – 1+ 0.2 xt – 2 + 4.1 + ut
where {ut} is a white noise time series with standard deviation = 2.0
White noise ≡ independent, mean zero (normal)
Find , (h), (h)
![Page 57: Time Series Analysis. Definition A Time Series {x t : t T} is a collection of random variables usually parameterized by 1) the real line T = R = (-∞,](https://reader035.vdocuments.us/reader035/viewer/2022062518/56649e525503460f94b480f7/html5/thumbnails/57.jpg)
1 21 1 1
1 22 1 1
To find (h) solve the equations:
or
1 (0.7)1 0.2 1
2 0.7 1 0.2 1
thus 0.7 0.71 0.875
1 .2 0.8
2 0.7 0.875 0.2 0.8125
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1 21 2h h h
for h > 2
This can be used in sequence to find:
0.7 1 0.2 2h h
3 , 4 , 5 , etc.
results
h 0 1 2 3 4 5 6 7 8h ) 1.0000 0.8750 0.8125 0.7438 0.6831 0.6269 0.5755 0.5282 0.4849 h
![Page 59: Time Series Analysis. Definition A Time Series {x t : t T} is a collection of random variables usually parameterized by 1) the real line T = R = (-∞,](https://reader035.vdocuments.us/reader035/viewer/2022062518/56649e525503460f94b480f7/html5/thumbnails/59.jpg)
pp
110
1
2
To find (0) use:
= 17.778
or
2
1 2
01 1 2
22.0
1 0.70 0.8750 0.20 0.8125
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0h h
To find (h) use:
1 21
4.1 4.1
411 0.70 0.20 0.1
To find use:
![Page 61: Time Series Analysis. Definition A Time Series {x t : t T} is a collection of random variables usually parameterized by 1) the real line T = R = (-∞,](https://reader035.vdocuments.us/reader035/viewer/2022062518/56649e525503460f94b480f7/html5/thumbnails/61.jpg)
An explicit formula for (h)
Auto-regressive time series of order p.
![Page 62: Time Series Analysis. Definition A Time Series {x t : t T} is a collection of random variables usually parameterized by 1) the real line T = R = (-∞,](https://reader035.vdocuments.us/reader035/viewer/2022062518/56649e525503460f94b480f7/html5/thumbnails/62.jpg)
Consider solving the difference equation:
011 phhh p
This difference equation can be solved by:Setting up the polynomial
pp xxx 11
pr
x
r
x
r
x111
21
where r1, r2, … , rp are the roots of the polynomial (x).
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The difference equation
011 phhh p
has the general solution:
h
pp
hh
rc
rc
rch
111
22
11
where c1, c2, … , cp are determined by using the starting values of the sequence (h).
![Page 64: Time Series Analysis. Definition A Time Series {x t : t T} is a collection of random variables usually parameterized by 1) the real line T = R = (-∞,](https://reader035.vdocuments.us/reader035/viewer/2022062518/56649e525503460f94b480f7/html5/thumbnails/64.jpg)
21
2
1
2
1110
ttt uxx 11
10
and
11 01
hhh 11 1 for h > 1
Example: An AR(1) time series
![Page 65: Time Series Analysis. Definition A Time Series {x t : t T} is a collection of random variables usually parameterized by 1) the real line T = R = (-∞,](https://reader035.vdocuments.us/reader035/viewer/2022062518/56649e525503460f94b480f7/html5/thumbnails/65.jpg)
The difference equation
011 hh Can also be solved by:Setting up the polynomial
xx 11
11
1
1 where1
r
r
x
Then a general formula for (h) is:
10 since 1
1111
1
hh
h
cr
ch
![Page 66: Time Series Analysis. Definition A Time Series {x t : t T} is a collection of random variables usually parameterized by 1) the real line T = R = (-∞,](https://reader035.vdocuments.us/reader035/viewer/2022062518/56649e525503460f94b480f7/html5/thumbnails/66.jpg)
tttt uxxx 2211
10
11 and 21
21 21 hhh
for h > 1
Example: An AR(2) time series
2
11 1
1or
![Page 67: Time Series Analysis. Definition A Time Series {x t : t T} is a collection of random variables usually parameterized by 1) the real line T = R = (-∞,](https://reader035.vdocuments.us/reader035/viewer/2022062518/56649e525503460f94b480f7/html5/thumbnails/67.jpg)
Setting up the polynomial
2211 xxx
2
1 2 1 2 1 2
1 1 11 1 1
x xx x
r r r r r r
2
22
111 2
4 where
r
2
22
112 2
4 and
r
212
211
1 and
11 :Note
rrrr
![Page 68: Time Series Analysis. Definition A Time Series {x t : t T} is a collection of random variables usually parameterized by 1) the real line T = R = (-∞,](https://reader035.vdocuments.us/reader035/viewer/2022062518/56649e525503460f94b480f7/html5/thumbnails/68.jpg)
Then a general formula for (h) is:
hh
rc
rch
22
11
11
For h = 0 and h = 1.
2
2
1
1
2
11 1
1r
c
r
c
211 cc
Solving for c1 and c2.
![Page 69: Time Series Analysis. Definition A Time Series {x t : t T} is a collection of random variables usually parameterized by 1) the real line T = R = (-∞,](https://reader035.vdocuments.us/reader035/viewer/2022062518/56649e525503460f94b480f7/html5/thumbnails/69.jpg)
Then a general formula for (h) is:
hh
rrrrr
rr
rrrrr
rrh
22121
212
12121
221 1
1
11
1
1
and
2121
221
1 1
1
rrrr
rrc
Solving for c1 and c2.
2121
212
2 1
1
rrrr
rrc
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If 04 22
1 21 and rr are real and
hh
rrrrr
rr
rrrrr
rrh
22121
212
12121
221 1
1
11
1
1
is a mixture of two exponentials
![Page 71: Time Series Analysis. Definition A Time Series {x t : t T} is a collection of random variables usually parameterized by 1) the real line T = R = (-∞,](https://reader035.vdocuments.us/reader035/viewer/2022062518/56649e525503460f94b480f7/html5/thumbnails/71.jpg)
If 04 22
1 21 and rr are complex conjugates.
ieRiyxr 1
ieRiyxr 2
2 2 1where and tan , tanx x
R x yy y
cos sin , cos sini ie i e i
cos , sin2 2
i i i ie e e e
i
Some important complex identities
![Page 72: Time Series Analysis. Definition A Time Series {x t : t T} is a collection of random variables usually parameterized by 1) the real line T = R = (-∞,](https://reader035.vdocuments.us/reader035/viewer/2022062518/56649e525503460f94b480f7/html5/thumbnails/72.jpg)
The above identities can be shown using the power series expansions:
2 3 4
12! 3! 4!
u u u ue u
2 4 6
cos 12! 4! 6!
u u uu
3 5 7
sin3! 5! 7!
u u uu u
![Page 73: Time Series Analysis. Definition A Time Series {x t : t T} is a collection of random variables usually parameterized by 1) the real line T = R = (-∞,](https://reader035.vdocuments.us/reader035/viewer/2022062518/56649e525503460f94b480f7/html5/thumbnails/73.jpg)
Some other trig identities:
1. cos cos cos sin sinu v u v u v
2. cos cos cos sin sinu v u v u v
3. sin sin cos cos sinu v u v u v
4. sin sin cos cos sinu v u v u v
2 25. cos 2 cos sinu u u
6. sin 2 2sin cosu u u
![Page 74: Time Series Analysis. Definition A Time Series {x t : t T} is a collection of random variables usually parameterized by 1) the real line T = R = (-∞,](https://reader035.vdocuments.us/reader035/viewer/2022062518/56649e525503460f94b480f7/html5/thumbnails/74.jpg)
ii
ii
eeRR
eReR
rrrr
rr
1
1
1
12
22
2121
221
ii
ii
eeRR
eReR
rrrr
rr
1
1
1
12
22
2121
212
sin212
2
iR
eRe ii
sin212
2
iR
eeR ii
![Page 75: Time Series Analysis. Definition A Time Series {x t : t T} is a collection of random variables usually parameterized by 1) the real line T = R = (-∞,](https://reader035.vdocuments.us/reader035/viewer/2022062518/56649e525503460f94b480f7/html5/thumbnails/75.jpg)
h
hiii
h
hiii
R
e
iR
eeR
R
e
iR
eRe
sin21sin21 2
2
2
2
Hence
hh
rrrrr
rr
rrrrr
rrh
22121
212
12121
221 1
1
11
1
1
sin212
11112
iRR
eeeeRh
hihihihi
sin1
1sin1sin2
2
RR
hhRh
![Page 76: Time Series Analysis. Definition A Time Series {x t : t T} is a collection of random variables usually parameterized by 1) the real line T = R = (-∞,](https://reader035.vdocuments.us/reader035/viewer/2022062518/56649e525503460f94b480f7/html5/thumbnails/76.jpg)
sin1
sincoscossinsincoscossin2
2
RR
hhhhRh
sin1
sincos1cossin12
22
RR
hRhRh
hR
hRR
h cotsin11
cos 2
2
hR
hh tansincos
cot1
1tan if
2
2
R
R
![Page 77: Time Series Analysis. Definition A Time Series {x t : t T} is a collection of random variables usually parameterized by 1) the real line T = R = (-∞,](https://reader035.vdocuments.us/reader035/viewer/2022062518/56649e525503460f94b480f7/html5/thumbnails/77.jpg)
hR
hD
cos
hR
hhh
tansincos Hence
hR
hh
sinsincoscos
cos
1
222
tan1cos
sincos
cos
1 where
D
a damped cosine wave
![Page 78: Time Series Analysis. Definition A Time Series {x t : t T} is a collection of random variables usually parameterized by 1) the real line T = R = (-∞,](https://reader035.vdocuments.us/reader035/viewer/2022062518/56649e525503460f94b480f7/html5/thumbnails/78.jpg)
Example
Consider the AR(2) time series:
xt = 0.7xt – 1+ 0.2 xt – 2 + 4.1 + ut
where {ut} is a white noise time series with standard deviation = 2.0
The correlation function found before using the difference equation:
(h) = 0.7 (h – 1) + 0.2 (h – 2)
h 0 1 2 3 4 5 6 7 8h ) 1.0000 0.8750 0.8125 0.7438 0.6831 0.6269 0.5755 0.5282 0.4849 h
![Page 79: Time Series Analysis. Definition A Time Series {x t : t T} is a collection of random variables usually parameterized by 1) the real line T = R = (-∞,](https://reader035.vdocuments.us/reader035/viewer/2022062518/56649e525503460f94b480f7/html5/thumbnails/79.jpg)
Alternatively setting up the polynomial
2 21 21 1 .7 .2x x x x x
1 2
1 1x x
r r
221 1 2
12
.7 .7 4 .24where
2 2 .2r
221 1 2
22
.7 .7 4 .24and
2 2 .2r
1.089454
4.58945
![Page 80: Time Series Analysis. Definition A Time Series {x t : t T} is a collection of random variables usually parameterized by 1) the real line T = R = (-∞,](https://reader035.vdocuments.us/reader035/viewer/2022062518/56649e525503460f94b480f7/html5/thumbnails/80.jpg)
Thus
hh
rrrrr
rr
rrrrr
rrh
22121
212
12121
221 1
1
11
1
1
1 2 1 21 22.7156r r r r
2 21 2 2 11 21.8578 and 1 0.85782r r r r
2 21 2 2 1
1 2 1 2 1 2 1 2
1 10.962237 and 0.037763
1 1
r r r r
r r r r r r r r
1 10.962237 0.037763
1.089454 4.58945
h h
h
![Page 81: Time Series Analysis. Definition A Time Series {x t : t T} is a collection of random variables usually parameterized by 1) the real line T = R = (-∞,](https://reader035.vdocuments.us/reader035/viewer/2022062518/56649e525503460f94b480f7/html5/thumbnails/81.jpg)
Another Example
Consider the AR(2) time series:
xt = 0.2xt – 1- 0.5 xt – 2 + 4.1 + ut
where {ut} is a white noise time series with standard deviation = 2.0
The correlation function found before using the difference equation:
(h) = 0.2 (h – 1) - 0.5 (h – 2)
h 0 1 2 3 4 5 6 7 8h ) 1.0000 0.8750 0.8125 0.7438 0.6831 0.6269 0.5755 0.5282 0.4849 h
![Page 82: Time Series Analysis. Definition A Time Series {x t : t T} is a collection of random variables usually parameterized by 1) the real line T = R = (-∞,](https://reader035.vdocuments.us/reader035/viewer/2022062518/56649e525503460f94b480f7/html5/thumbnails/82.jpg)
Alternatively setting up the polynomial
2 21 21 1 .2 .5x x x x x
1 2
1 1x x
r r
221 1 2
12
.2 .2 4 0.54where
2 2 0.5r
221 1 2
22
.2 .2 4 0.54and
2 2 0.5r
.2 1.96.2 1.96
1i
.2 1.96.2 1.96
1i
![Page 83: Time Series Analysis. Definition A Time Series {x t : t T} is a collection of random variables usually parameterized by 1) the real line T = R = (-∞,](https://reader035.vdocuments.us/reader035/viewer/2022062518/56649e525503460f94b480f7/html5/thumbnails/83.jpg)
Thus
1 .2 1.96 ir i R e
2 .2 1.96 ir i R e
where
2 2 2.2 1.96 2R x y
and0.2
tan 0.142857,1.96
x
y
1thus tan 0.142857 0.141897
![Page 84: Time Series Analysis. Definition A Time Series {x t : t T} is a collection of random variables usually parameterized by 1) the real line T = R = (-∞,](https://reader035.vdocuments.us/reader035/viewer/2022062518/56649e525503460f94b480f7/html5/thumbnails/84.jpg)
2
2
1 2 1Now tan cot cot .14897 2.33333
1 2 1
R
R
1Thus tan 2.33333 1.165905
2
2.538591cos 0.141897 1.165905
2h
h
cosFinally
h
D hh
R
2 2Also 1 tan 1 2.3333 2.538591D
![Page 85: Time Series Analysis. Definition A Time Series {x t : t T} is a collection of random variables usually parameterized by 1) the real line T = R = (-∞,](https://reader035.vdocuments.us/reader035/viewer/2022062518/56649e525503460f94b480f7/html5/thumbnails/85.jpg)
hR
hD
cos
hR
hhh
tansincos Hence
hR
hh
sinsincoscos
cos
1
222
tan1cos
sincos
cos
1 where
D
a damped cosine wave
![Page 86: Time Series Analysis. Definition A Time Series {x t : t T} is a collection of random variables usually parameterized by 1) the real line T = R = (-∞,](https://reader035.vdocuments.us/reader035/viewer/2022062518/56649e525503460f94b480f7/html5/thumbnails/86.jpg)
Conditions for stationarity
Autoregressive Time series of order p, AR(p)
![Page 87: Time Series Analysis. Definition A Time Series {x t : t T} is a collection of random variables usually parameterized by 1) the real line T = R = (-∞,](https://reader035.vdocuments.us/reader035/viewer/2022062518/56649e525503460f94b480f7/html5/thumbnails/87.jpg)
The value of xt increases in magnitude and ut eventually becomes negligible.
i.e. 11 ttt uxx
If 1 = 1 and = 0.
The time series {xt|t T} satisfies the equation:
The time series {xt|t T} exhibits deterministic behaviour.
11 tt xx
![Page 88: Time Series Analysis. Definition A Time Series {x t : t T} is a collection of random variables usually parameterized by 1) the real line T = R = (-∞,](https://reader035.vdocuments.us/reader035/viewer/2022062518/56649e525503460f94b480f7/html5/thumbnails/88.jpg)
Let 1, 2, … p denote p numbers.
Let {ut|t T} denote a white noise time series with variance 2.
– independent– mean 0, variance 2.
Let {xt|t T} be defined by the equation.
2211 tptpttt uxxxx
Then {xt|t T} is called a Autoregressive time series of order p. AR(p)
![Page 89: Time Series Analysis. Definition A Time Series {x t : t T} is a collection of random variables usually parameterized by 1) the real line T = R = (-∞,](https://reader035.vdocuments.us/reader035/viewer/2022062518/56649e525503460f94b480f7/html5/thumbnails/89.jpg)
Consider the polynomial
pp xxx 11
pr
x
r
x
r
x111
21
with roots r1, r2 , … , rp
then {xt|t T} is stationary if |ri| > 1 for all i.
If |ri| < 1 for at least one i then {xt|t T} exhibits deterministic behaviour.
If |ri| ≥ 1 and |ri| = 1 for at least one i then {xt|t T} exhibits non-stationary random behaviour.
![Page 90: Time Series Analysis. Definition A Time Series {x t : t T} is a collection of random variables usually parameterized by 1) the real line T = R = (-∞,](https://reader035.vdocuments.us/reader035/viewer/2022062518/56649e525503460f94b480f7/html5/thumbnails/90.jpg)
Special Cases: The AR(1) time
Let {xt|t T} be defined by the equation.
11 ttt uxx
![Page 91: Time Series Analysis. Definition A Time Series {x t : t T} is a collection of random variables usually parameterized by 1) the real line T = R = (-∞,](https://reader035.vdocuments.us/reader035/viewer/2022062518/56649e525503460f94b480f7/html5/thumbnails/91.jpg)
Consider the polynomial
xx 11
1
1r
x
with root r1= 1/1
1. {xt|t T} is stationary if |r1| > 1 or |1| < 1 .
2. If |ri| < 1 or |1| > 1 then {xt|t T} exhibits deterministic behaviour.
3. If |ri| = 1 or |1| = 1 then {xt|t T} exhibits non-stationary random behaviour.
![Page 92: Time Series Analysis. Definition A Time Series {x t : t T} is a collection of random variables usually parameterized by 1) the real line T = R = (-∞,](https://reader035.vdocuments.us/reader035/viewer/2022062518/56649e525503460f94b480f7/html5/thumbnails/92.jpg)
Special Cases: The AR(2) time
Let {xt|t T} be defined by the equation.
2211 tttt uxxx
![Page 93: Time Series Analysis. Definition A Time Series {x t : t T} is a collection of random variables usually parameterized by 1) the real line T = R = (-∞,](https://reader035.vdocuments.us/reader035/viewer/2022062518/56649e525503460f94b480f7/html5/thumbnails/93.jpg)
Consider the polynomial
2211 xxx
21
11r
x
r
x
where r1 and r2 are the roots of (x)
1. {xt|t T} is stationary if |r1| > 1 and |r2| > 1 .
2. If |ri| < 1 or |1| > 1 then {xt|t T} exhibits deterministic behaviour.
3. If |ri| ≤ 1 for i = 1,2 and |ri| = 1 for at least on i then {xt|t T} exhibits non-stationary random behaviour.
This is true if 1+2 < 1 , 2 –1 < 1 and 2 > -1.These inequalities define a triangular region for 1 and 2.
![Page 94: Time Series Analysis. Definition A Time Series {x t : t T} is a collection of random variables usually parameterized by 1) the real line T = R = (-∞,](https://reader035.vdocuments.us/reader035/viewer/2022062518/56649e525503460f94b480f7/html5/thumbnails/94.jpg)
Patterns of the ACF and PACF of AR(2) Time SeriesIn the shaded region the roots of the AR operator are complex
h kk
h kk
h kk
h kk
1
21
-1
2-2
III
IIIIV
2