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Outline
�Modeling objectives in time series
�General features of ecological/environmental time series
�Components of a time series
�Frequency domain analysis-the spectrum
�Estimating and removing seasonal components
�Other cyclical components
�Putting it all together
Introduction to Statistical Analysis of Time SeriesRichard A. Davis
Department of Statistics
2
Time Series: A collection of observations xt, each one being recorded at time t. (Time could be discrete, t = 1,2,3,…, or continuous t > 0.)
Objective of Time Series Analaysis�Data compression
-provide compact description of the data.�Explanatory
-seasonal factors-relationships with other variables (temperature, humidity, pollution, etc)
�Signal processing-extracting a signal in the presence of noise
�Prediction-use the model to predict future values of the time series
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General features of ecological/environmental time seriesExamples.
1. Mauna Loa (CO2,, Oct `58-Sept `90)C
O2
1960 1970 1980 1990
320
330
340
350
Features
� increasing trend (linear, quadratic?)
� seasonal (monthly) effect.
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Go to ITSM Demo
2. Ave-max monthly temp (vegetation=tundra,, 1895-1993)
Features
� seasonal (monthly effect)
� more variability in Jan than in July
tem
p
0 200 400 600 800 1000 1200
-50
510
1520
5
tem
p
0 20 40 60 80 100
-50
510
1520
July: mean = 21.95, var = .6305
Jan : mean = -.486, var =2.637
tem
p
0 20 40 60 80 100
1516
1718
1920
Sept : mean = 17.25, var =1.466Line: 16.83 + .00845 t
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Components of a time series
Classical decompositionXt = mt + st + Yt
• mt = trend component (slowly changing in time)
• st = seasonal component (known period d=24(hourly), d=12(monthly))
• Yt = random noise component (might contain irregularcyclical components of unknown frequency + other stuff).
Go to ITSM Demo
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Estimation of the components.
Xt = mt + st + Yt
12/)5.5(.ˆ 6556 ++−− ++++= ttttt xxxxm �
kkt tataam +++= �10ˆ
� polynomial fitting
Trend mt
� filtering. E.g., for monthly data use
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Estimation of the components (cont).
Xt = mt + st + Yt
years ofnumber ,/)(ˆ 2412 =++= ++ NNxxxs tttt �
)122sin()
122cos(ˆ tBtAst
π+π=
Irregular component Yt
tttt smXY ˆˆˆ −−=
Seasonal st
� Use seasonal (monthly) averages after detrending. (standardize so that st sums to 0 across the year.
� harmonic components fit to the time series using least squares.
9
Toy example. (n=6)
/sqrt(3))' )66
2(cos ,),6
2(cos(1ππ= …c
/sqrt(3))' )6622(cos ,),
622(cos(2
ππ= …c
/sqrt(6)' )622(cos ,),
22(cos(3
ππ= …c
/sqrt(6))' )66
02(cos ,),6
02(cos(0ππ= …c
/sqrt(3))' )66
2(sin ,),6
2(sin(1ππ= …s
/sqrt(3))' )66
22(sin ,),6
22(sin(2ππ= …s
c0c1
s1c2
s2c3
x
X=(4.24, 3.26, -3.14, -3.24, 0.739, 3.04)’X=(4.24, 3.26, -3.14, -3.24, 0.739, 3.04)’ = 2c0+5(c1+s1)-1.5(c2+s2)+.5c3
The spectrum and frequency domain analysis
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Fact: Any vector of 6 numbers, x = (x1, . . . , x6)’ can be written as a linear combination of the vectors c0, c1, c2, s1, s2, c3.
More generally, any time series x = (x1, . . . , xn)’ of length n (assume n is odd) can be written as a linear combination of the basis (orthonormal) vectors c0, c1, c2, …, c[n/2], s1, s2, …, s[n/2]. That is,
]2/[ ,111100 nmbabaa mmmm =++++= scsccx �
ω
ωω
=
ω
ωω
=
=
)sin(
)2sin()sin(
2 ,
)cos(
)2cos()cos(
2 ,
1
11
1 2/12/12/1
0
j
j
j
j
j
j
j
j
nn
nnn ���
scc
11
Properties:
1. The set of coefficients {a0, a1, b1, … } is called the discrete Fourier transform
]2/[ ,111100 nmbabaa mmmm =++++= scsccx �
∑
∑
∑
=
=
=
ω==
ω==
==
n
tjtjj
n
tjtjj
n
tt
txn
b
txn
a
xn
a
12/1
2/11
2/1
2/11
2/100
)sin(2),(
)cos(2),(
1),(
sx
cx
cx
12
( )∑∑==
++=m
jj
n
tt baax
1j
2220
1
2
2. Sum of squares.
3. ANOVA (analysis of variance table)
Source DF Sum of Squares Periodgram
ω0 1 a02 I(ω0)
ω1=2π/n 2 a12 + b1
2 2 I(ω1)
ωm =2πm/n 2 am2 + bm
2 2 I(ωm)
n
� ��
∑t
tx2
�
13
Source DF Sum of Squares
ω0=0 (period 0) 1 a02 = 4.0
ω1=2π/6 (period 6) 2 a12 + b1
2 = 50.0
ω2 =2π2/6 (period 3) 2 a22 + b2
2 = 4.5
ω3 =2π3/6 (period 2) 1 a32 = 0.25
6 = 58.75
Applied to toy example
∑t
tx2
Test that period 6 is significant
H0: Xt = µ + εt , {εt} ~ independent noise
H1: Xt = µ + A cos (t2π/6) + B sin (t2π/6) + εt
Test Statistic: (n-3)I(ω1)/(Σt xt2-I(0)-2I(ω1)) ~ F(2,n-3)
(6-3)(50/2)/(58.75-4-50)=15.79 ⇒ p-value = .003
14
Ex. Sinusoid with period 12.
.120,,2,1 ),122sin(3)
122cos(5 …=π+π= tttxt
ITSM DEMO
The spectrum and frequency domain analysis
Ex. Sinusoid with periods 4 and 12.
Ex. Mauna Loa
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Sometimes, a seasonal component with period 12 in the time series can be removed by differencing at lag 12. That is the differenced series is
.120,,2,1 ,)122sin(3)
122cos(5 …=ε+π+π= tttx tt
Differencing at lag 12
12−−= ttt xxy
Now suppose xt is the sinusoid with period 12 + noise.
Then
which has correlation at lag 12.
1212 −− ε−ε=−= ttttt xxy
16
Ex. Sunspots.
� period ~ 2π/.62684=10.02 years
� Fisher’s test ⇒ significance
What model should we use?
ITSM DEMO
Other cyclical components; searching for hidden cycles
17
Noise.The time series {Xt} is white or independent noise if the sequence of random variables is independent and identically distributed.
time
x_t
0 20 40 60 80 100 120
-20
24
Battery of tests for checking whiteness.In ITSM, choose statistics => residual analysis => Tests of Randomness
18
Residuals from Mauna Loa data.
x_t
x_{t+
1}
-1.0 -0.5 0.0 0.5 1.0 1.5
-1.0
-0.5
0.0
0.5
1.0
1.5 Cor(Xt, Xt+1) = .824
t rt rt+1
1 -.19 -.142 -.14 -.253 -.25 -.134 -.13 -.32
…
x_t
x_{t+
1}
-1.0 -0.5 0.0 0.5 1.0 1.5
-1.0
-0.5
0.0
0.5
1.0
1.5 Cor(Xt, Xt+2) = .736
t rt rt+2
1 -.19 -.252 -.14 -.133 -.25 -.324 -.13 -.02
…
t rt rt+1
1 -.19 -.132 -.14 -.323 -.25 -.134 -.13 -.02
…
x_t
x_{t+
1}
-1.0 -0.5 0.0 0.5 1.0 1.5
-1.0
-0.5
0.0
0.5
1.0
1.5 Cor(Xt, Xt+3) = .654
x_t
x_{t+
1}
-1.0 -0.5 0.0 0.5 1.0 1.5
-1.0
-0.5
0.0
0.5
1.0
1.5 Cor(Xt, Xt+25) = .074 t rt rt+25
1 -.19 .132 -.14 .043 -.25 .204 -.13 .47
…
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lag
ACF
0 10 20 30 40
0.0
0.2
0.4
0.6
0.8
1.0
Autocorrelation function (ACF):
Mauna Loa residuals
lag
ACF
0 10 20 30 40
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
white noise
Conf Bds: ± 1.96/sqrt(n)
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Putting it all together
Example: Mauna LoaC
O2
1960 1970 1980 1990
320
340
trend
1960 1970 1980 1990
320
340
seas
onal
1960 1970 1980 1990
-3-1
12
3
irreg
ular
par
t
1960 1970 1980 1990
-1.0
0.0
1.0
21
Strategies for modeling the irregular part {Yt}.� Fit an autoregressive process� Fit a moving average process� Fit an ARMA (autoregressive-moving average) process
In ITSM, choose the best fitting AR or ARMA using the menu option
Model => Estimation => Preliminary => AR estimationor
Model => Estimation => Autofit
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How well does the model fit the data?1. Inspection of residuals.
Are they compatible with white (independent) noise?� no discernible trend� no seasonal component� variability does not change in time.� no correlation in residuals or squares of residuals
Are they normally distributed?
2. How well does the model predict.� values within the series (in-sample forecasting)� future values
3. How well do the simulated values from the model capture the characteristics in the observed data?
ITSM DEMO with Mauna Loa
23
Model refinement and Simulation� Residual analysis can often lead to model refinement� Do simulated realizations reflect the key features
present in the original data
Two examples� Sunspots� NEE (Net ecosystem exchange).
Limitations of existing models� Seasonal components are fixed from year to year.� Stationary through the seasons� Add intervention components (forest fires, volcanic
eruptions, etc.)
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Other directions� Structural model formulation for trend and seasonal
components� Local level model
mt = mt-1 + noiset
� Seasonal component with noise
st = – st-1 – st-2 – . . . – st-11+ noiset
� Xt= mt + st + Yt + εt
� Easy to add intervention terms in the above formulation.
� Periodic models (allows more flexibility in modeling transitions from one season to the next).
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