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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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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


  • 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

Esterby-IMS Jan17,2008


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

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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

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First example:

Niagara River at Niagara-on-the-Lake

monthly means

1976 to 1992

1. Total phosphorus (TP)

2. Nitrate nitrogen

3. (Discharge )

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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)

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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

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Nonparametric methods

Context

Data bases:short temporal records

many variables measured

many stations

Objective:assess temporal changes in water quality

Notation

(yij,tij,xij) yij pwater quality indicators

xijqcovariates

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 .

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Detection and Estimation

Detection

  • Mann-Kendall statistic

  • Seasonal Kendall trend test

  • Heterogeneity

  • Serial Correlation

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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)

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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)

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Estimation of trend

Theil-Sen slope estimator

Slope estimator, Bj, for season j

median of the n(n- 1)/2 quantities ( ykj − yij)/(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 ( ykj − yij)

for i =1,2,…,c and k=c+1,…,n

or median over all seasons

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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 ν1=n1-p-1, ν2=n2-q-1 and n2=n- n1

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

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Lake Erie monitoring stations

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Change in log productivity, going from south shore to north shore of the Lake Erie

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Change in log productivity, going from east to west in the Lake Erie

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Relative marginal likelihood for n1 and the fitted regression lines with the pollen concentration plotted versus depth in the sediment core

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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

Esterby-IMS Jan17,2008


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

Esterby-IMS Jan17,2008


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)

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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

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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?

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Cleveland, R. B.. Cleveland, W. S., McRae, J. E., and Terpenning, I. 1990. ‘STL: A seasonal-trend decomposition procedure based on loess’, J. OffStat., 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.

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