1. Time series analysis - lecture 5
Dynamic regression with exponentially decreasing coefficients in the transfer function
Consider the dynamic regression model
where Yt = the forecast variable (output series);
Xt = the explanatory variable (input series);
Nt = an ARIMA-process representing the combined effect of all other factors influencing Yt
and
(B) = ( 1+ B + 2B2 + … + kBk)
If k is large, then
Time series analysis - lecture 5

2. Parsimonious parameterization of dynamic regression models
where Yt = the forecast variable (output series);
Xt = the explanatory variable (input series);
Nt = an ARIMA-process representing the combined effect of all other factors influencing Yt
Important special cases: s = 0 and r = 0
Time series analysis - lecture 5

3. Time series analysis - lecture 5
Selecting the order of a general dynamic regression model - values to be determined
We need to determine:
the values of r, s, and b
the values of p, d, and q in the ARIMA(p, d, q) model of the noise Nt
the values of P, D, and Q of a seasonal ARIMA model, if such data are analysed
Time series analysis - lecture 5

4. Time series analysis - lecture 5
Selecting the order of a general dynamic regression model - LTF identification
Step 1: Fit a multiple regression model with a low order AR model for the noise
Step 2: If the errors from the regression appear to be non-stationary, then difference Y and X
Step 3: Determine b, r, and s by inspecting the -weights in the regression
Step 4: Compute the series of noise terms and fit an ARMA-model to these terms
Step 5: Refit the entire model using the new ARMA model for the noise
Time series analysis - lecture 5

5. Proc ARIMA in SAS – stationary inputs and outputs
proc arima data=timeseri.gasfurnace;
identify var=CO2 crosscorr=gasrate;
estimate p=1 input=(3$/(1)gasrate);
run;
Time series analysis - lecture 5

6. Proc ARIMA in SAS - nonstationary inputs and outputs
proc arima data=timeseri.gasfurnace;
identify var=CO2 (1) crosscorr=gasrate(1);
estimate p=1 input=(3$/(1)gasrate);
run;
Time series analysis - lecture 5

7. Intervention analysis
where Yt = the forecast variable (output series);
Xt = a step or pulse function;
Nt = the combined effect of all other factors influencing Yt (the noise);
(B) = ( 0 +  1B +  2B2 + … +  kBk), where k is the order of the transfer function
Time series analysis - lecture 5

8. Intervention analysis - delayed or decayed response
Delayed response:
Xt is a step function
Decayed response:
Xt is a pulse function
Time series analysis - lecture 5

9. Statistical methods for trend detection
Concordant pair
Linear regression
Smoothing techniques
Non-parametric tests
Discordant pair
Concordant pair
Time series analysis - lecture 5

10. Time series analysis - lecture 5
The classical Mann-Kendall test for monotone trend in a single time series of data
Test statistic:
where
Under H0, the test statistic T is approximately normal with mean zero and variance one
Basic idea:
Consider all pairs (Xi, Xj) of observations.
Subtract the number of “discordant pairs” from the number of “concordant pairs”
Time series analysis - lecture 5

11. Time series analysis - lecture 5
Mann-Kendall tests for monotone trends in data collected over several seasons Correction for short-term serial dependence
Hirsch&Slack (1984):
Compute a Mann-Kendall statistic Tj for each season j=1,…,m and form
T=T1+…+Tm
Derive the variance of T by using the Dietz-Killeen estimator of the covariance matrix of (T1,…,Tm)
Year Q1 Q2 Q3 Q4
1996
5.5
4.3
3.5
4.6
1997
6.8
3.7
4.9
1998
3.1
2.6
2.1
1999
3.0
2.4
3.4
2000
2.5
1.9
2.7
2001
2.8
2.3
2002
2003
2004
Test
statistic T1 T2 T3 T4
Time series analysis - lecture 5

12. Time series analysis - lecture 5
Mann-Kendall tests for monotone trends in data collected at several sites Correction for dependence between sites
Loftis et al. (1991):
Compute a Mann-Kendall statistic Tj for each site j=1,…,m and form
T=T1+…+Tm
Derive the variance of T by using the Dietz-Killeen estimator of the covariance matrix of (T1,…,Tm)
Year
Stn1
Stn2
Stn3
Stn4
1996
5.5
4.3
3.5
4.6
1997
6.8
3.7
4.9
1998
3.1
2.6
2.1
1999
3.0
2.4
3.4
2000
2.5
1.9
2.7
2001
2.8
2.3
2002
2003
2004
Test
statistic T1 T2 T3 T4
Time series analysis - lecture 5

13. From multiple time series of data to a smooth trend surface
Time series analysis - lecture 5

14. Time series analysis - lecture 5
A simple model for simultaneous smoothing and adjustment for a single covariate
Let be the observed response for the jth coordinate the ith year,
and let denote a contemporaneous value of a covariate. Assume that .
Response
Deterministic
trend
Impact of covariate
Random error
Time series analysis - lecture 5

15. Time series analysis - lecture 5
A semiparametric model for simultaneous smoothing and adjustment for several covariates
Let be the observed response for the jth class the ith year,
and let denote contemporaneous values of covariates.
Assume that .
Random error
Response
Deterministic
trend
Impact of covariate
Impact of covariate
Time series analysis - lecture 5

16. Gradient smoothing Penalty of irregular interannual variation.
Penalty of irregular interannual variation
Penalty of irregular variation along the gradient
Time series analysis - lecture 5

17. Spatial smoothing along a gradient Temporal smoothing across years
Smoothing of the trend function in models of time series data representing several sites along a gradient
Spatial smoothing along a gradient
Temporal smoothing across years
Time series analysis - lecture 5

18. Sequential smoothing across seasons Temporal smoothing across years
Smoothing of the trend function in models of time series data representing several seasons
Sequential smoothing across seasons
Temporal smoothing across years
Time series analysis - lecture 5

19. Time series analysis - lecture 5
Smoothing of the trend function in models of time series data representing several sectors
Circular smoothing
across sectors
Temporal smoothing
across years
Time series analysis - lecture 5

20. Total phosphorus concentrations at Dagskärsgrund in Lake Vänern
Time series analysis - lecture 5

21. Time series analysis - lecture 5
Detection of an unknown level shift - the Standard Normal Homogeneity Test
Let X1, …, Xn be a sequence of independent normal random variables with variance one
H0: E(Xj) = 0 for 1  j  n
H1: E(Xj) = 1 for j  m
E(Xj) = 2 for j > m
Test statistic
where and are the average responses before and after the shift
Time series analysis - lecture 5

22. Detection of a level shift and change-point detection
Time series analysis - lecture 5

23. Time series analysis - lecture 5
Total phosphorus concentrations at Dagskärsgrund in Lake Vänern -simultaneous smoothing and change-point detection
Time series analysis - lecture 5

24. Time series analysis - lecture 5
Ratio between TOC (total organic carbon) and permanganate consumption in Swedish rivers
Time series analysis - lecture 5

25. Time series analysis - lecture 5
Ratio between TOC (total organic carbon) and permanganate consumption in Swedish rivers - simultaneous smoothing and change-point detection
Time series analysis - lecture 5

26. Time series analysis - lecture 5
Uncertainty assessment and hypothesis testing in joint models of smooth trends and change-points
Resampling techniques are needed
Standard resampling techniques must be modified to accommodate correlated residuals
Time series analysis - lecture 5