1. Time series modelling and statistical trends
Marian Scott and Adrian Bowman
SEPA, July 2012

2. smoothing a time series
In many time series, the seasonal variation can be so strong that it obscures any trend or cyclical component.
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

3. 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

4. smoothing a time series
We have data , where Xt = number of bus passengers on the t'th day. Since the periodic variation is repeated every 7 days, a 7-period moving-average (Mt) is used to smooth the series, where:
Mt = 1/7{Xt-3+Xt-2+…..+Xt+3}
This averages out the seasonality, since each Mt is an average over 7 different 'seasons' (days of the week). Note, though, that Mt is only defined for t = 4, 5, ..., N-7.
other smoothing methods (more modern) commonly used include Lowess

5. smoothing a time series
one of the most commonly used smoothing techniques is moving average.
smooth series: Yi = wkYi+k
3-point, 5-point, 7-point moving average example
window may be chosen to reflect the periodicity of the data series
other smoothing methods (more modern) commonly used include Lowess

6. 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.

7. water surface temperature from Jan 1981- Feb 1992 (Piegorsch)- with lowess curve

8. Example : different smoothing technique applied to air quality data (that have been logged)

9. 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?
This allows us to capture the regular repeat pattern in each year – the seasonal effect

10. Yi = 0 +  cos (2[ti - ]/p) + i
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 +  cos (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

11. Yi = 0 +  sin (2[ti - ]/p) + i
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(2ti/p) and si = sin(2ti/p)
So a regression model

12. Example: red curve shows the harmonic pattern (superimposed on a declining trend).

13. Example to try Qn 4 in practical3final.txt
The script shows how we can create the new explanatory variables, doy is a new variable that records where in the year (which day from 366) the sample was taken.

14. seasonal indices and de-seasonalisation
The reason for giving the seasonally-adjusted data is to make trends and cycles more apparent.
seasonal adjustments best explained as
step 1: define the Yt =Xt –Mt (actual-smoothed)
step 2: average all the Yt values for each ‘season’ to give the same seasonal index (e.g. for quarterly data there would be 4 values), S
step 3: the seasonally adjusted data Xt- S

15. 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)
Data are NOT independent

16. 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)- as the correlation between original series and a version shifted back k time units
horizontal lines show approximate 95% confidence intervals for individual coefficients.

17. Example: ACF of raw water temperature data

18. 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 called ARIMA).
how should we remove the seasonal pattern or the trend?

19. 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

20. Example: ACF of water temperature data

21. Example 1: ACF of water temperature data- difference order 12

22. Examples to try In practical4.txt Exercises 1 and 2
Why is correlation important
How good is the ACF as a diagnostic
Exercise 2 shows the output from a single command stl (which is a decomposition of the data series into trend, seasonal component and residual)

23. 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) decomposition

24. An example for you to try
Exercise 3, Central England temperature
obtain the acf
use the stl() command
Look at monthly data

25. A different type of change
Change can be
Abrupt
As a result of an intervention
So we might like to consider a slightly different form of model

26. Nile flow
relatively poor fit of straight line model, lots of variation.
some pattern in the residuals

27. A straight line model for the Nile
relatively poor fit, lot of variation.
any pattern in the residuals?
this residual plot is against order of the observations

28. a non-parametric model for the Nile
a smooth function (LOESS) or non-parametric regression model
Seem OK?
any suggestion that there may be a change-point?

29. A different type of change
So we might like to consider a slightly different form of model- the river Nile was dammed in the late 1800s
So there may be a changepoint- a shift in the mean flow level, and if so can we see it.

30. the smooths
Two smooth curves are fit and we identify the biggest discrepancy between then
with confidence bands added, helps identify the change location
Where is the biggest discrepancy?

31. An alternative model for the Nile
two smooth sections, broken at roughly 1900.
different mean levels in the two periods
so modelling the two periods separately

32. The moral of this example
Trends can be challenging to identify
Modelling needs to be flexible
We need to be mindful of the assumptions

33. An example
Haddocks- this is an example about fish stocks, we can try fitting some very simple time series regression models.
We might want to predict what fish stocks might be several years in the future