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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:
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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
Climate change over time Trend analysis
The concern is:
Use numbers to draw conclusions
Trends in means, although variability and extremes are important
As applied to water quality, but consider relevance to topics of
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
Niagara River at Niagara-on-the-Lake
1976 to 1992
1. Total phosphorus (TP)
2. Nitrate nitrogen
3. (Discharge )
Data bases: short temporal records
many variables measured
Objective: assess temporal changes in water quality
(yij,tij,xij) yij pwater quality indicators
tijday of the jth sample collection in year i
one water quality indicator, one covariate and monthly sampling
(yij, tij, xij)for j = 1,2,. . . , 12, i = 1 , 2,. . . , nand tij= 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)
(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)
Theil-Sen slope estimator
Slope estimator, Bj, for season j
median of the n(n- 1)/2 quantities ( ykj − yij)/(k −i)
for i or B, median over all seasons Hodges-Lehman estimator Step change at c, for season j median of all differences ( ykj − yij) for i =1,2,…,c and k=c+1,…,n or median over all seasons Esterby-IMS Jan17,2008
or B, median over all seasons
Step change at c, for season j
median of all differences ( ykj − yij)
for i =1,2,…,c and k=c+1,…,n
or median over all seasons
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 ν1=n1-p-1, ν2=n2-q-1 and n2=n- n1
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
A number of ways to do this
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
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
ie. relevance to data sets used with models
Trying to predict change by modelling processes, do we have evidence?
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.