1. Trend analysis: considerations for water quality management Sylvia R. Esterby Mathematics, Statistics and Physics, University of British Columbia Okanagan Kelowna BC Canada Week 2 January 14-18 of: Data-driven and Physically-based Models for Characterization of Processes in Hydrology, Hydraulics, Oceanography and Climate Change Institute for Mathematical Sciences, National University of Singapore January 7-28, 2008

2. Esterby-IMS Jan17,20082 Introduction Type of water quality data considered Accounting for heterogeneity Nonparametric methods Analogous regression methods Decomposing series Many stations Homogeneity over time and space in parameter estimation for data-driven models

3. Esterby-IMS Jan17,20083 Introduction Climate change over time  Trend analysis The concern is: Pollutants increasing Response variables are changing Use numbers to draw conclusions Model generated, variable of direct interest Observed/measured, variable of direct interest Observed/measured, proxy variable Trends in means, although variability and extremes are important As applied to water quality, but consider relevance to topics of workshop

4. Esterby-IMS Jan17,20084 Water quality First consideration is heterogeneity other than that of primary interest (heterogeneity exists or we are finished once we “calculate the mean”) Most important to consider here is seasonal cycle Two ways of doing this: - Block on season - Decompose series into components for trend, season and residual View data in way that corresponds to way we model variability in the data

5. Esterby-IMS Jan17,20085 First example: Niagara River at Niagara-on-the-Lake monthly means 1976 to 1992 1. Total phosphorus (TP) 2. Nitrate nitrogen 3. (Discharge )

6. Esterby-IMS Jan17,20086

7. 7 Left: Annual seasonal cycle TP, monthly mean for each year plotted against month. Right: Change over years for TP displayed for each month (read across and then down)

8. Esterby-IMS Jan17,20088 Mean monthly total phosphorus, TP, (mg/L, solid line) and discharge (dashed line, m/s) in the Niagara River at Niagara-on-the-Lake, 1976 to 1992

9. Esterby-IMS Jan17,20089 Nonparametric methods Context Data bases:short temporal records many variables measured many stations Objective:assess temporal changes in water quality Notation (y ij, t ij,x ij ) y ij p water quality indicators x ij q covariates t ij day of the jth sample collection in year i one water quality indicator, one covariate and monthly sampling (y ij, t ij, x ij ) for j = 1,2,..., 12, i = 1, 2,..., n and t ij = j.

10. Esterby-IMS Jan17,200810 Detection and Estimation Detection Mann-Kendall statistic Seasonal Kendall trend test Heterogeneity Serial Correlation

11. Esterby-IMS Jan17,200811 The Mann-Kendall statistic for season j sgn(x) =−1 if x < 0 0 if x = 0 1 if x > 0 Hypothesis: random sample of n iid variables. (powerful for departures in the form of monotonic change over time) Seasonal Kendall trend test (Hirsch et al., 1982)

12. Esterby-IMS Jan17,200812 Decompose gives tests of heterogeneity and trend (van Belle and Hughes, 1984) Assumption of independence within season tenable Modifications for serial correlation of observations within year Dietz and Killeen (1981), El-Shaarawi and Niculescu (1992),others Covariates (eg. Remove effect of flow and use adjusted values)

13. Esterby-IMS Jan17,200813 Estimation of trend Theil-Sen slope estimator Slope estimator, B j, for season j median of the n(n - 1)/2 quantities ( y kj − y ij )/(k −i) for i<k and i,k=1,2,…,n or B, median over all seasons Hodges-Lehman estimator Step change at c, for season j median of all differences ( y kj − y ij ) for i =1,2,…,c and k=c+1,…,n or median over all seasons

14. Esterby-IMS Jan17,200814 Parametric analogues Linear and polynomial regression with seasons as blocks Same change for each season Estimation of point of change in regression model Marginal maximum likelihood estimator for time of change Esterby and El-Shaarawi(1981), El-Shaarawi and Esterby(1982) polynomials of degree p, q determine ν 1 =n 1 -p-1, ν 2 =n 2 -q-1 and n 2 =n- n 1 Two examples 1. Lake Erie (courtesy El-Shaarawi). Primary productivity in Lake Erie: - changes south to north - changes east to west 2. Proxy variable for time in the past, Ambrosia pollen horizon

15. Esterby-IMS Jan17,200815

16. Esterby-IMS Jan17,200816 Lake Erie monitoring stations

17. Esterby-IMS Jan17,200817 Change in log productivity, going from south shore to north shore of the Lake Erie

18. Esterby-IMS Jan17,200818 Change in log productivity, going from east to west in the Lake Erie

19. Esterby-IMS Jan17,200819 Relative marginal likelihood for n 1 and the fitted regression lines with the pollen concentration plotted versus depth in the sediment core

20. Esterby-IMS Jan17,200820 Decomposing series A number of ways to do this Regression could add more terms to seasonal component dependent or independent errors Smoothing with LOESS or STL seasonal trend decomposition procedure based on LOESS (Cleveland et al, 1990), generalized additive modelling with splines Example smoothing of nitrate nitrogen in Niagara River

21. Esterby-IMS Jan17,200821 Mean monthly total phosphorus, TP, (mg/L, solid line) and discharge (dashed line, m/s) in the Niagara River at Niagara-on-the-Lake, 1976 to 1992

22. Esterby-IMS Jan17,200822 Decomposition of nitrate nitrogen in the Niagara River using smoothing for trend, loess smoothing of the residuals from trend, and residuals from trend and seasonal components (data are shown in top plot)

23. Esterby-IMS Jan17,200823 Many stations interest in change/no change at each station often summarize conclusion graphically or in summaries Could use tests: nonparametric extensions, test parameters in regression, homogeneity of curves

24. Esterby-IMS Jan17,200824 Homogeneity over time and space in parameter estimation for data-driven models ie. relevance to data sets used with models Trying to predict change by modelling processes, do we have evidence?

25. Esterby-IMS Jan17,200825 Cleveland, R. B.. Cleveland, W. S., McRae, J. E., and Terpenning, I. 1990. ‘STL: A seasonal- trend decomposition procedure based on loess’, J. Off Stat., 6, 3-73. Cleveland, W. S., and Grosse, E. 1991. ‘Computational methods for local regression’, Statistics in Computing, 1, 47-62. Dietz, E. J.. and Killeen, T. J. 198 1. ‘A non-parametric multivariate test for monotone trend with pharmaceutical applications’, J. Am. Stat. Assoc., 76, 169-174. El-Shaarawi, A. H., and Niculescu, S. 1992. ‘On Kendall’s tau as a test for trend in time series data’, Environmetrics. 3, 385-41 I. Esterby, S.R. 1996. ‘Review of methods for the detection and estimation of trends with emphasis on water quality applications’, Hydrological Processes, 10, 127-149. Esterby, S. R. 1993. ’Trend analysis methods for environmental data’, Environmetrics. 4, 459- 481. Esterby, S. R.. and El-Shaarawi, A. H. 1981. ‘Inference about the point of change in a regression model’, Appl. Statis., 30, 277-285. Hirsch, R. M., Slack, J. R., and Smith, R. A. 1982. ‘Techniques of trend analysis for monthly water quality data’, Wat. Resour. Res., 18, 107-121. van Belle, G., and Hughes, J. P. 1984. ‘Nonparametric tests for trend in water quality’, Wat. Resour. Res., 20, 127-136.