Means Recall: We model a time series as a collection of random - - PowerPoint PPT Presentation
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Means Recall: We model a time series as a collection of random - - PowerPoint PPT Presentation
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Means Recall: We model a time series as a collection of random variables: x 1 , x 2 , x 3 , . . . , or more generally { x t , t T } . The mean function is x,t = E( x t ) = xf t ( x ) dx where the expectation is for the
SLIDE 1
Means
Recall: We model a time series as a collection of random
variables: x1, x2, x3, . . . , or more generally {xt, t ∈ T }.
The mean function is
µx,t = E(xt) =
∞
∞ xft(x)dx
where the expectation is for the given t, across all the possible values of xt. Here ft(·) is the pdf of xt.
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Example: Moving Average
wt is white noise, with E (wt) = 0 for all t
the moving average is
vt = 1 3
wt−1 + wt + wt+1
so
µv,t = E (vt) = 1 3
E
wt−1 + E (wt) + E
wt+1
= 0.
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Moving Average Model with Mean Function
Time v 100 200 300 400 500 −2 −1 1
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Example: Random Walk with Drift
The random walk with drift δ is
xt = δt +
t
j=1
wj
so
µx,t = E (xt) = δt +
t
j=1
E
wj
= δt,
a straight line with slope δ.
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Random Walk Model with Mean Function
Time x 100 200 300 400 500 20 40 60 80
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Example: Signal Plus Noise
The “signal plus noise” model is
xt = 2 cos(2πt/50 + 0.6π) + wt
so
µx,t = E (xt) = 2 cos(2πt/50 + 0.6π) + E (wt) = 2 cos(2πt/50 + 0.6π), the (cosine wave) signal.
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Signal-Plus-Noise Model with Mean Function
Time x 100 200 300 400 500 −4 −2 2 4
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Covariances
The autocovariance function is, for all s and t,
γx(s, t) = E
(xs − µx,s)
xt − µx,t
.
Symmetry: γx(s, t) = γx(t, s).
Smoothness:
– if a series is smooth, nearby values will be very similar, hence the autocovariance will be large; – conversely, for a “choppy” series, even nearby values may be nearly uncorrelated.
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Example: White Noise
If wt is white noise wn(0, σ2
w), then
γw(s, t) = E (wswt) =
σ2
w,
s = t, 0, s = t.
definitely choppy!
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Autocovariances of White Noise
s t gamma
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Example: Moving Average
The moving average is
vt = 1 3
wt−1 + wt + wt+1
and E (vt) = 0, so
γv(s, t) = E (vsvt) = 1 9E
ws−1 + ws + ws+1
wt−1 + wt + wt+1
=
(3/9)σ2
w,
s = t (2/9)σ2
w,
s = t ± 1 (1/9)σ2
w,
s = t ± 2 0,
therwise.
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Autocovariances of Moving Average
s t gamma
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Example: Random Walk
The random walk with zero drift is
xt =
t
j=1
wj and E (xt) = 0
so
γx(s, t) = E (xsxt) = E
s
j=1
wj
t
j=1
wj
= min{s, t}σ2
w.
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Autocovariances of Random Walk
s t gamma
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Notes:
– For the first two models, γx(s, t) depends on s and t only through |s − t|, but for the random walk γx(s, t) depends
n s and t separately.
– For the first two models, the variance γx(t, t) is constant, but for the random walk γx(t, t) = tσ2
w increases indefi-
nitely as t increases.
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Correlations
The autocorrelation function (ACF) is
ρ(s, t) = γ(s, t)
γ(s, s)γ(t, t)
Measures the linear predictability of xt given only xs.
Like any correlation, −1 ≤ ρ(s, t) ≤ 1.
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Across Series
For a pair of time series xt and yt, the cross covariance
function is γx,y(s, t) = E
(xs − µx,s)
yt − µy,t
.
The cross correlation function (CCF) is
ρx,y(s, t) = γx,y(s, t)
γx(s, s)γy(t, t)
.
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Stationary Time Series
Basic idea: the statistical properties of the observations do
not change over time.
Two specific forms: strong (or strict) stationarity and weak
stationarity.
A time series xt is strongly stationary if the joint distribution
f every collection of values {xt1, xt2, . . . , xtk} is the same as
that of the time-shifted values {xt1+h, xt2+h, . . . , xtk+h}, for every dimension k and shift h.
Strong stationarity is hard to verify.
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If {xt} is strongly stationary, then for instance:
k = 1: the distribution of xt is the same as that of xt+h, for
any h; – in particular, if we take h = −t, the distribution of xt is the same as that of x0; – that is, every xt has the same distribution;
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k = 2: the joint (bivariate) distribution of (xs, xt) is the same
as that of (xs+h, xt+h), for any h; – in particular, if we take h = −t, the joint distribution of (xs, xt) is the same as that of (xs−t, x0); – that is, the joint distribution of (xs, xt) depends on s and t only through s − t;
and so on...
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A time series xt is weakly stationary if:
– the mean function µt is constant; that is, every xt has the same mean; – the autocovariance function γ(s, t) depends on s and t only through their difference |s − t|.
Weak stationarity depends only on the first and second mo-
ment functions, so is also called second-order stationarity.
