time series modelling marian scott sages, march 2009
TRANSCRIPT
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Time series modelling
Marian Scott
SAGES, March 2009
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what is a time series?
• a time series is a sequence of measurements made over time.
• notationally, this would commonly be written as y1, y2,…, yi, ….yT
• the index i denotes the position in the sequence of observations
• for this early session, we will assume that the data are equally spaced-so that i is truly an index
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how to plot the data
a time series plot• choice of the x-axis scale
– occasionally, each observation is indexed by its position in the sequence (OK if equally spaced)
– alternatively, we may use the actual timescale (e.g. if an annual series, years or a daily series, then days 1-365)
– or we may regard time on a continuous scale (time might be recorded in decimal form e.g. 1986.5- which would be June 1986)
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How is biodiversity changing (EEA CSI 009)
• Populations of common and widespread farmland bird species in 2003 are only 71% of their 1980 levels.
• an annual indicator
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Water quality- freshwater (CSI 020)
• Concentrations of P generally decreased
• Nitrate concentrations have remained constant
• What are the rates of change and are they significant?
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Example 1- monthly mean CO2 levels
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daily mean temperature
days
de
gre
e c
elc
ius
-10
01
02
0
01/01/1973 01/01/1980 01/01/1987 12/31/1993 12/31/2000
daily minima temperature
days
de
gre
e c
elc
ius
-20
-10
01
0
01/01/1973 01/01/1980 01/01/1987 12/31/1993 12/31/2000
Example 2: a time series plot (daily values)
the x-axis shows the actual date
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Example 3- Some typical environmental series- Loch Leven
(NERC-CEH)SRP
Years
SR
P,
mu
g/l
1970 1980 1990 2000
02
04
06
0
TP
Years
TP
, m
ug
/l
1970 1980 1990 2000
50
10
01
50
Secchi
Years
Se
cch
i, m
etr
es
1970 1980 1990 2000
12
34
Daphnia
Years
Da
ph
nia
, in
div
idu
als
/l
1970 1980 1990 2000
02
04
06
08
0
Chlorophyll
Years
Ch
loro
ph
yll,
mu
g/l
1970 1980 1990 2000
05
01
00
15
02
00
Water Temperature
Years
Wa
ter
Te
mp
era
ture
, o
C
1970 1980 1990 20000
51
01
52
0
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SO2 monitored in AT02
observation number
ug
S/m
3
0 2000 4000 6000 8000
02
04
06
08
01
00
Example 4- air quality, monitored through time (from EMEP programme)
note the gaps and the rather extreme values- one strategy is to take logs
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Time series data features
• patterns over time (both short and long term)
• often missing data- may cause problems for statistical analysis
• variation, which may not be constant over time so may need to consider transformations (log)
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Seasonal patterns (cycles)
• in many environmental times series, we could imagine some periodicity (e.g. such as a monthly pattern in temperature)
• so it is common to produce a “seasonality plot”
• the index (x-axis scale) depends on the period over which the cycle repeats itself.
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days
resid
ua
ls
0
-20
2
smoothing residuals of SO2 Vs days of the year with a bandwidth of 30, GB02, p.value = 0
0 30 60 90 120 150 180 210 240 270 300 330 360
days
resid
ua
ls
0
-4-2
02
smoothing residuals of SO2 Vs days of the year with a bandwidth of 30, AT02, p.value = 0
0 30 60 90 120 150 180 210 240 270 300 330 360
Example 1: daily observations, so the seasonal curve is plotted over days of the year
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SunSatFriThuWedTueMon
0.04
0.02
0.00
-0.02
-0.04
-0.06
Seasonal Indices
15
10
5
0
Percent Variation, by Seasonal Period
43210
-1-2-3-4
Original Data, by Seasonal Period
43210
-1-2-3-4
Residuals, by Seasonal Period
Seasonal Analysis for ln(SO2) monitored in GB02
Mon
MonMon
Tue
TueTue
Wed
WedWed
Thu
Thu ThuFri Fri
Fri Sat
SatSat Sun
Sun
Sun
Example 2: Daily data- data are plotted over the days of the week
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Log SRP
Month
Lo
g S
RP
, m
ug
/l
2 4 6 8 10 12
-20
24
Log TP
MonthL
og
TP
, m
ug
/l2 4 6 8 10 12
3.5
4.0
4.5
5.0
Log Chlorophyll
Month
Lo
g C
hlo
rop
hyl
l, m
ug
/l
2 4 6 8 10 12
01
23
45
Log Daphnia
Month
Lo
g D
ap
hn
ia,
ind
ivid
ua
ls/l
2 4 6 8 10 12
-4-2
02
4
Log Secchi
Month
Lo
g S
ecc
hi,
me
tre
s
2 4 6 8 10 12
-0.5
0.0
0.5
1.0
Water Temperature
Month
Wa
ter
Tem
pe
ratu
re,
oC
2 4 6 8 10 12
05
10
15
20
Example 3: Loch Leven, monthly data- data are plotted over the months of the year (Lowess smooth included)
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what are the questions of interest?
• we want to know about trends, where a trend is defined to be:– the long-term sweep of the data.
• we want to know about possible seasonality (or cycles)– The seasonal component of a time series
describes a regular fluctuation which has a period. (The period is the time interval between consecutive peaks or troughs.)
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a descriptive model
• A useful descriptive model for a time series consists of 3 components:
• X = Trend + Seasonal Component + Irregular Component
or X = T+S+I• I is the irregular component, which is left over
when the trend, and seasonal components are all accounted for. It is an irregular or random fluctuation (like residuals in regression).
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smoothing a time series
• In many time series, the seasonal variation can be so strong that it obscures any trend or cyclical component. However, for understanding the process being observed (and forecasting future values of the series), trends and cycles are of prime importance. Smoothing is a process designed to remove seasonality so that the long-term movements in a time series can be seen more clearly
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smoothing a time series
• one of the most commonly used smoothing techniques is moving average.
• difficult choice: the window over which to smooth
• smooth series: Yi = wkYi+k
• other smoothing methods (more modern) commonly used include Lowess
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smoothing a time series
• LO(W)ESS, is a method that is known as locally weighted polynomial regression. At each point in the data set a low-degree polynomial is fit to a subset of the data, with explanatory variable values near the point whose response is being estimated. The polynomial is fit using weighted least squares, giving more weight to points near the point whose response is being estimated and less weight to points further away.
• Many of the details of this method, such as the degree of the polynomial model and the weights, are flexible.
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Example 1: water surface temperature from Jan 1981- Feb 1992 (Piegorsch)- with lowess curve
19/12/
1991
01/11/
1990
27/09/
1989
24/08/19
88
21/0
7/19
87
19/0
6/19
86
17/0
5/19
85
12/0
4/19
84
10/0
3/19
83
09/0
2/19
82
20/0
1/19
81
30
25
20
15
10
5
0
date
tem
p
Time Series Plot of temp
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Example 1: water surface temperature -seasonal pattern
121086420
30
25
20
15
10
5
0
month
tem
pScatterplot of temp vs month
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Example 1: water surface temperature- seasonal pattern by week
6050403020100
30
25
20
15
10
5
0
week
tem
pScatterplot of temp vs week
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Example 1: water surface temperature- variability by year
199219911990198919881987198619851984198319821981
30
25
20
15
10
5
0
year
tem
pBoxplot of temp
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Example 1: water surface temperature-variability by month
121110987654321
30
25
20
15
10
5
0
month
tem
p
Boxplot of temp
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Example 1: water surface temperature-moving average length 52
19/12/19
91
01/1
1/19
90
27/0
9/19
89
24/08/
1988
21/07/
1987
19/06/
1986
17/0
5/19
85
12/0
4/19
84
10/0
3/19
83
09/02/
1982
20/01/
1981
30
25
20
15
10
5
0
date
tem
p
Length 52Moving Average
MAPE 44.8212MAD 6.1001MSD 48.9017
Accuracy Measures
ActualFits
Variable
Moving Average Plot for temp
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days
ln(u
g S
/m3
)
0
-4-3
-2-1
01
23
45
a) smoothing of the logarithm of SO2bandwidth = 30
0 730 1460 2190 2920 3650 4380 5110 5840 6570 7300 8030
days
ln(u
g S
/m3
)
0
-4-3
-2-1
01
23
45
b) smoothing of the logarithm of SO2bandwidth = 800
0 730 1460 2190 2920 3650 4380 5110 5840 6570 7300 8030
Example 2: different smoothing technique applied to air quality data (that have been logged)
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harmonic regression
• another way of a) describing and b) hence being able to remove the periodic component is to use what is called harmonic regression
• remember sin and cos from school?
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harmonic regression
• build a regression model using the sine function. sin () lies between -1 and +1, where measured in radians.
• for a periodic time series Yi we can build a regression model
• Yi = 0 + sin (2[ti - ]/p) + i
• to make this simpler, if we assume that p is known, this can be written as a simple multiple linear regression model
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harmonic regression
• for a periodic time series Yi we can build a regression model
• Yi = 0 + sin (2[ti - ]/p) + i
• to make this simpler,
• Yi = 0 + 1ci + 2si + i
• where ci = cos(2ti/p) and si = sin(2ti/p)
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ln(SO2) in GB02 against fine gridModel 2
weeks
ln(u
g S
/m3
)
1980 1985 1990 1995 2000
-2-1
01
23
Example 2: red curve shows the harmonic pattern (superimposed on a declining trend).
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correlation through time
• in many situations, we expect successive observations to show correlation at adjacent time points (most likely stronger the closer the time points are), strength of dependence usually depends on time separation or lag
• for regularly spaced data, we typically make use of the autocorrelation function (ACF)
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correlation through time
• for regularly spaced time series, with no missing data, we define the sample mean in the usual way
• then the sample autocorrelation coefficient at lag k ( 0), r(k)
• correlation between original series and a version shifted back k time units
• horizontal lines show approximate 95% confidence intervals for individual coefficients.
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Example 1: ACF of water temperature data
605550454035302520151051
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
Lag
Auto
corr
ela
tion
Autocorrelation Function for temp(with 5% significance limits for the autocorrelations)
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correlation through time
• ACF shows a very marked cyclical pattern• interpretation of the ACF
– we need to have removed both trend and seasonality– we hope that (for simplicity in subsequent modelling)
that only a few correlation coefficients (at small lags) will be significant.
• ACF an important diagnostic tool for time series modelling (formal models ARIMA). Formal time series models …see later session on trends
• how should we remove the seasonal pattern or the trend?
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differencing
• a common way of removing a simple trend (eg linear) is by differencing
• define a new series
• Zt = Yt – Yt-1
• a common way of removing seasonality (if we know the period to be p), is to take pth differences
• Zt = Yt – Yt-p
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Example 1: ACF of water temperature data
302520151051
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
Lag
Auto
corr
ela
tion
Autocorrelation Function for monthlymean(with 5% significance limits for the autocorrelations)
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Example 1: ACF of water temperature data- difference order 12
30282624222018161412108642
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
Lag
Auto
corr
ela
tion
Autocorrelation Function for 12 difference of monthly mean(with 5% significance limits for the autocorrelations)
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a descriptive model
• A useful descriptive model for a time series consists of 3 components:
• X = Trend + Seasonal Component + Irregular Component
or X = T+S+I• I is the irregular component, which is left over
when the trend and seasonal components are all accounted for. It is an irregular or random fluctuation (like residuals in regression).
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simple algorithm
• obtain rough estimate of trend (smoothing but one not affected by seasonality):
• subtract estimated trend• estimate seasonal cycle from detrended
series• what is left is the irregular component, • good alternative- STL (seasonal trend
lowess) decompostion (stl() command in R)
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a couple of examples for you to try
• for monthly temperature data– obtain the acf – use the stl() command
• for dissolved oxygen in River Clyde – fit a seasonal regression model
• In the final session on trend detection we will return to regression for time series.