Trend analysis: considerations for water quality management Sylvia R. Esterby Mathematics, Statistics and Physics, University of British Columbia Okanagan.

Slides:



Advertisements
Similar presentations
Detection of Hydrological Changes – Nonparametric Approaches
Advertisements

P ROBLEMS IN DETECTING TREND IN HYDROMETEOROLOGICAL SERIES FOR CLIMATE CHANGE STUDIES Jasna Plavšić 1 and Zoran Obušković 2 1 University of Belgrade –
Introduction to modelling extremes
Introduction to modelling extremes Marian Scott (with thanks to Clive Anderson, Trevor Hoey) NERC August 2009.
Environmental change and statistical trends – some examples
Measurement and assessment of change What it the status quo in environmental science? In time – A simple trend line – A p-value or a 95% confidence interval.
Introduction Describe what panel data is and the reasons for using it in this format Assess the importance of fixed and random effects Examine the Hausman.
Statistical modelling of precipitation time series including probability assessments of extreme events Silke Trömel and Christian-D. Schönwiese Institute.
1 McGill University Department of Civil Engineering and Applied Mechanics Montreal, Quebec, Canada.
Hierarchical Linear Modeling: An Introduction & Applications in Organizational Research Michael C. Rodriguez.
Combining Information from Related Regressions Duke University Machine Learning Group Presented by Kai Ni Apr. 27, 2007 F. Dominici, G. Parmigiani, K.
Chapter 4 Randomized Blocks, Latin Squares, and Related Designs
Budapest May 27, 2008 Unifying mixed linear models and the MASH algorithm for breakpoint detection and correction Anders Grimvall, Sackmone Sirisack, Agne.
Derwent R.G. et al., 1998, “Observation and interpretation of the seasonal cycles in the surface concentration of ozone and carbon monoxide at Mace Head,
Extremes ● An extreme value is an unusually large – or small – magnitude. ● Extreme value analysis (EVA) has as objective to quantify the stochastic behavior.
Some more issues of time series analysis Time series regression with modelling of error terms In a time series regression model the error terms are tentatively.
Time series analysis - lecture 5
Extreme Value Analysis, August 15-19, Bayesian analysis of extremes in hydrology A powerful tool for knowledge integration and uncertainties assessment.
Quantitative Business Analysis for Decision Making Simple Linear Regression.
Fault Prediction and Software Aging
Autocorrelation Lecture 18 Lecture 18.
Business Statistics - QBM117 Statistical inference for regression.
Water Quality Monitoring and Parameter Load Estimations in Lake Conway Point Remove Watershed and L’Anguille River Watershed Presented by: Dan DeVun, Equilibrium.
Hydrologic Statistics
Simple Linear Regression
Comparing Two Samples Harry R. Erwin, PhD
Modeling errors in physical activity data Sarah Nusser Department of Statistics and Center for Survey Statistics and Methodology Iowa State University.
Spatial Interpolation of monthly precipitation by Kriging method
Interim Update: Preliminary Analyses of Excursions in the A.R.M. Loxahatchee National Wildlife Refuge August 18, 2009 Prepared by SFWMD and FDEP as part.
1 Trend Analysis Step vs. monotonic trends; approaches to trend testing; trend tests with and without exogeneous variables; dealing with seasonality; Introduction.
Model Comparison for Tree Resin Dose Effect On Termites Lianfen Qian Florida Atlantic University Co-author: Soyoung Ryu, University of Washington.
Water Quality Monitoring and Constituent Load Estimation in the Kings River near Berryville, Arkansas 2009 Brian E. Haggard Arkansas Water Resources Center.
Determining homogenous regions: considerations for water quality management Sylvia R. Esterby Mathematics, Statistics and Physics University of British.
The Regional Kendall Test for Trend Dennis Helsel US Geological Survey.
Week 11 Introduction A time series is an ordered sequence of observations. The ordering of the observations is usually through time, but may also be taken.
Statistics and Modelling 3.1 Credits: 3 Internally Assessed.
SADC Course in Statistics Forecasting and Review (Sessions 04&05)
Introduction 1. Climate – Variations in temperature and precipitation are now predictable with a reasonable accuracy with lead times of up to a year (
1 …continued… Part III. Performing the Research 3 Initial Research 4 Research Approaches 5 Hypotheses 6 Data Collection 7 Data Analysis.
Computational statistics, lecture3 Resampling and the bootstrap  Generating random processes  The bootstrap  Some examples of bootstrap techniques.
Business Statistics for Managerial Decision Farideh Dehkordi-Vakil.
Based on the Mezentsev-Choudhury-Yang equation (with n representing catchments characteristics): and water balance equation R = P ─ E, Yang et al. [2011]
WISKI for Hydrologic Assessment Michael Seneka Water Policy Branch Alberta Environment and Sustainable Resource Development.
MANAGEMENT SCIENCE AN INTRODUCTION TO
Identification of Extreme Climate by Extreme Value Theory Approach
Sea Level in month i =α 2 sin t i + α 3 cos t i + α 4 sin2 t i + α 5 cos2 t i seasonality + α 6 SOI i climate indices + α 1 + α 7 time i + (deviations.
Psychology 202a Advanced Psychological Statistics October 22, 2015.
STATISTICS OF EXTREME EVENTS AMS Probability and Statistics Committee January 11, 2009 WELCOME! AMS SHORT COURSE.
The Mixed Effects Model - Introduction In many situations, one of the factors of interest will have its levels chosen because they are of specific interest.
Stats Term Test 4 Solutions. c) d) An alternative solution is to use the probability mass function and.
1 Detection of discontinuities using an approach based on regression models and application to benchmark temperature by Lucie Vincent Climate Research.
QM Spring 2002 Business Statistics Analysis of Time Series Data: an Introduction.
Tim Cohn USGS Office of Surface Water Reston, Virginia Flood Frequency Analysis in Context of Climate Change.
Water Quality Sampling, Analysis and Annual Load Determinations for the Illinois River at Arkansas Highway 59 Bridge, 2008 Brian E. Haggard Arkansas Water.
Testing for equal variance Scale family: Y = sX G(x) = P(sX ≤ x) = F(x/s) To compute inverse, let y = G(x) = F(x/s) so x/s = F -1 (y) x = G -1 (y) = sF.
Chapter 20 Time Series Analysis and Forecasting. Introduction Any variable that is measured over time in sequential order is called a time series. We.
Running and jumping Time and space records: long jump, one hundred meters are getting closer. (NG)
1 Optimizing sampling methods for pollutant loads and trends in San Francisco Bay urban stormwater monitoring Aroon Melwani, Michelle Lent, Ben Greenfield,
Introduction Many problems in Engineering, Management, Health Sciences and other Sciences involve exploring the relationships between two or more variables.
Actions & Activities Report PP8 – Potsdam Institute for Climate Impact Research, Germany 2.1Compilation of Meteorological Observations, 2.2Analysis of.
of Temperature in the San Francisco Bay Area
Chapter 12 Trend Analysis
Analysis of Hydro-climatology of Malawi
Effects of Climate Change on the Great Lakes
Determining Trends in Values of Concern During a Specified Time Period
of Temperature in the San Francisco Bay Area
Statistics in WR: Lecture 22
Simple Linear Regression - Introduction
ENM 310 Design of Experiments and Regression Analysis Chapter 3
Trend assessment (A. V, 2.4.4) Identification of trends in pollutants
Presentation transcript:

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 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

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

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

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

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

Esterby-IMS Jan17,20086

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)

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

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.

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

Esterby-IMS Jan17, 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)

Esterby-IMS Jan17, 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)

Esterby-IMS Jan17, 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

Esterby-IMS Jan17, 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

Esterby-IMS Jan17,200815

Esterby-IMS Jan17, Lake Erie monitoring stations

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

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

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

Esterby-IMS Jan17, 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, 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, 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)

Esterby-IMS Jan17, 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

Esterby-IMS Jan17, 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?

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