1. A generic procedure for simultaneous estimation of monotone trends and seasonal patterns in time series of environmental data by Mohamed Hussian and Anders Grimvall mohus@mai.liu.se

2. Monday 16-June- 20032 Outline l Introduction and aim of study l Monotone regression in one independent variable l The surface response problem  Formulation of the Problem  Algorithm  Results l Examples of monotone relationships in environmental data l Monotone regression in two independent variables l Conclusions

3. Monday 16-June- 20033 Introduction l The decomposition of a time series of data into  a trend curve,  a set of seasonal components  and irregular variation is one of the classical problems in time series analysis. l The presently used decomposition methods are primary developed for economic time series of data and official statistics l Time series of environmental data has the following characteristics.  the seasonal pattern is smooth and has usually one major peak each year.  monotone trends are of specific interest.

4. Monday 16-June- 20034 Tot-N concentrations (mg/l) Monthly mean concentrations of total nitrogen at Brunsbüttel in the Elbe River Year Month

5. Monday 16-June- 20035 Aim of study l The aim is to develop procedures that enable simultaneous estimation of monotone trends and relatively smoothly varying seasonal components.  Can be done by presenting decomposition methods that can accommodate the (prior information) constraints on the extracted components.

6. Monday 16-June- 20036 Tot-N concentrations (Brunsbuttel) in the Elbe River Mean values for July 1985-2000

7. Monday 16-June- 20037 Monotone regression l Given a set of two-dimensional data  Sort the data by X into  Minimize under the constraints A well-known algorithm used to solve the problem is the PAV Algorithm (Pool- Adjacent-Violators Algorithm), (Barlow et al., 1972; Hanson et al., 1973)

8. Monday 16-June- 20038 The PAV Algorithm Start with, move to the right and stop if (, ) violates the monotonicity constraint. Pool and the adjacent by replacing them both with their average, = = ( + )/2 Check if . If not, pool ,,  into one average. Continue to the left until the monotonicity requirement is satisfied. Proceed to the right. The final solutions are x y y (1) y (2) y (3) y (i-1) y (i+1) y (i)(i) y y (n)(n)

9. Monday 16-June- 20039 l Facts  If the data are already monotone, then PAVA will reproduce them.  The solution is a step function.  If there are outliers or aberrant observations, then PAVA will produce long, flat levels. l Two different strategies using the PAVA  Smooth the data and then search for a monotone approximation of the smooth using the PAVA (Friedman and Tibshirani, 1984).  Apply the PAVA and then smooth the data (Hildenbrand, 1986). The PAV Algorithm cont ´

10. Monday 16-June- 200310 The problem formulation l denote a time series of data collected over m seasons denote the sum of the trend and seasonal components at time i l can be determined by minimizing under the following constraints

11. Monday 16-June- 200311 The problem formulation cont ´ l Monotonicity Constraints is either decreasing or increasing for each season l Seasonality Constraints how the function is composed of convex and concave curve pieces, l for all time points belonging to a given season.

12. Monday 16-June- 200312 An example of linear trend with a superimposed trigonometric seasonal pattern satisfying the given constraints

13. Monday 16-June- 200313 Algorithm General Information l The problem is a classical quadratic optimization problem. l The computational burden increases rapidly with the number of variables and constraints. l This burden can be a serious problem if the suggested algorithms do not take into considerations the special features of the constraints

14. Monday 16-June- 200314 Algorithm cont ´ Theoretical Solution l Given a crude initial estimate of l Form new estimates, k = 1, 2, …, by employing an updating formula : is a vector defining the shape of the adjustment h : is a scaling factor

15. Monday 16-June- 200315 l h is determined in such a way that l is minimised and the desired constraints are satisfied. l Applying such a solution will reduce the original multivariate optimisation problem to a sequence of univariate optimisation problems. Algorithm cont´

16. Monday 16-June- 200316 Response surface satisfying monotonicity and convexity constraints

17. Monday 16-June- 200317 Results l The algorithm described above has performed satisfactorily on water quality data from the Elbe River and other rivers l Regardless of the features of the data sets that were examined, the obtained sequences of fitted surfaces converges to a function that could be interpreted as a sum of trend and seasonal components l The extraction of residuals facilitated the detection of outliers l More work to be done to optimise the suggested algorithm to be much faster.

18. Monday 16-June- 200318 Estimation of trends and seasonal components in the presence of covariates l This problem can often be solved by making isotonic regression in two independent variables in the presence of seasonal components l The idea is to combine an existing algorithm for isotonic regression in two independent variables based on the PAV algorithms with specific partial ordering of the independent variables ( Dykstra et al., 1984) with the seasonal constraints.

19. Monday 16-June- 200319 Average monthly ozone concentrations versus humidity at Ähtäri in central Finland

20. Monday 16-June- 200320 Tot-P concentrations (Brunsbuttel) versus water discharge(NeuDarchau) in the Elbe River Mean values for April 1985-2000

21. Monday 16-June- 200321 ab Monthly Tot-P concentrations versus a) oxygen b) salinity monitored at the sampling site BY10 in the Baltic proper

22. Monday 16-June- 200322 Monotone regression in two independent variables l An algorithm for computing the least squares regression function which is constrained to be nondecreasing in each of several independent variables was developed by R. Dykstra & T. Robertson, 1984.  The algorithm was written specifically for two independent variables, and it is to produce the solution of  where is a given two-dimensional array of the original values; is a nonnegative array of weights; and K is the class of two-dimensional arrays, G=( ) such that  whenever

23. Monday 16-June- 200323  The algorithm is iterative in nature and uses successive one-dimensional smoothing by the use of the PAV Algorithm.  The problem of the order of smoothing ( rows and then columns or columns and then rows) has an influence on the number of iteration the algorithm takes to converge,  it does two cycles starting with rows and two cycles starting with columns, then it compares the total change between the values after that. It begins the smoothing process in the order corresponding to the smaller change Monotone regression with two independent variables cont´

24. Monday 16-June- 200324 Conclusions l The proposed decomposition method is just one example of a large class of smoothing procedures that could be,  further developed for quality assessment and routine analysis of environmental data collected over several seasons. l Further research can be done to,  Extend the Algorithm to deal with dynamic peaks instead of fixed peak each year  Using the suggested procedure, normalisation methods based on linear statistical models can be improved where we assume that the input of the nutrients to the sea is a monotone function of a number of forcing variables.