Regression Analysis of Phosphorous Loading Data for the Maumee River, Water Years 2003-2005 Charlie...
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![Page 1: Regression Analysis of Phosphorous Loading Data for the Maumee River, Water Years 2003-2005 Charlie Piette David Dolan Pete Richards Department of Natural.](https://reader036.fdocuments.us/reader036/viewer/2022062714/56649d695503460f94a47a20/html5/thumbnails/1.jpg)
Regression Analysis of Phosphorous Loading Data for
the Maumee River, Water Years 2003-2005
Charlie Piette
David Dolan
Pete Richards
Department of Natural and Applied Sciences
University of Wisconsin Green Bay
National Center for Water Quality Research, Heidelberg College
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Phosphorus and the Great Lakes Water Quality Agreement
• Goal for reduction
• Initial targets
• Secondary targets
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Maumee River Watershed
5
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Maumee River Facts
• Size
• Contribution
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Data Source• USGS
• NCWQR
• Used data from
WY 2003-2005
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Purpose of Our Research
• ECOFORE 2006: Hypoxia Assessment in Lake Erie
• Estimate TP loads to Lake Erie using data from Heidelberg College and effluent data from permitted point sources
• Constructing a daily time series of phosphorus loading (Maumee River)
![Page 10: Regression Analysis of Phosphorous Loading Data for the Maumee River, Water Years 2003-2005 Charlie Piette David Dolan Pete Richards Department of Natural.](https://reader036.fdocuments.us/reader036/viewer/2022062714/56649d695503460f94a47a20/html5/thumbnails/10.jpg)
Problems in Constructing a Time Series for the Maumee
• Missing data
• All three years missing some data
• No major precipitation events were missed in water years 2003 and 2004
• 2005……..
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Water Year 2005 Data Overview
• Missing an important time period
• December 2004-January 2005, moving the lab
• Very significant period of precipitation
• 32.8 inches of snow in January ’05
• Third wettest January on record
• Warm temps- 52˚F on New Year’s Day
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Importance of WY 2005
• Fifth largest peak flow in 73 year data record- 94,100 cfs
• Orders of magnitude larger than average flows for the same time period in WY ’03 and ’04
• 3,437cfs and 10,039 cfs respectively
• Need to model the missing data to complete the time series
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Objectives• Use statistical analysis to develop a model
for predicting missing T.P. for the Maumee in WY 2005
• Calculate an annual load for WY 2005 using measured and predicted data
• Compare estimated regression load to estimated load from another method
• Assess effectiveness of final regression model on other Lake Erie Tributaries
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Reconstructing the Missing Concentration Data
• Multiple regression w/ SAS
• Producing an equation that can be used to model for the missing phosphorus concentrations
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Basic Regression Equation• Y=ßо + ß1X1 + ß2X2 + ……… ßpXp + E• The terms…..
- 3. 0
- 2. 5
- 2. 0
- 1. 5
- 1. 0
- 0. 5
0. 0
LnFl ow
5. 0 5. 5 6. 0 6. 5 7. 0 7. 5 8. 0 8. 5 9. 0 9. 5 10. 0 10. 5 11. 0
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Basic Assumption of Regression• Linear relationship between dependent
and independent variables
- 3. 5
- 3. 0
- 2. 5
- 2. 0
- 1. 5
- 1. 0
- 0. 5
0. 0
LnFl ow
4 5 6 7 8 9 10 11 12
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Basic Assumptions: Continued• Normal distribution of residuals
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So, the data is suitable for regression analysis. What makes for a strong model?
• Hypothesis for model significance
• Hypothesis for parameter estimate significance
• P-values- <.05
• R2 value
• M.S.E.
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Beale’s Equation
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Beale’s Ratio Estimator• Daily load for
sampled days• Mean daily load• Flow-adjusted mean
daily load• Bias-corrected• X 365 = annual load
estimate
Date Flow P_Concentration
10/1/2003 10644.720 0.346
10/2/2003 7858.308 .
10/3/2003 5656.312 0.300
10/4/2003 4195.272 0.239
10/5/2003 2974.260 0.226
10/6/2003 2629.872 0.207
10/7/2003 2222.868 0.181
10/8/2003 1961.968 0.174
10/9/2003 1909.788 0.163
10/10/2003 1377.552 .
10/11/2003 1116.652 .
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Beale Stratified Ratio Estimator• Stratification- flow or time
• More accurate estimation
• “It’s an art!”
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Beale Vs. Regression• Both a means to the same end- annual
load estimate
• Both relying on one main assumption- a linear relationship
• Big difference- Beale is not good for reconstructing a time series
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Regression Analysis
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Data Analysis Step 1
• Transforming the data to log space
0. 0
0. 1
0. 2
0. 3
0. 4
0. 5
0. 6
0. 7
0. 8
Fl ow
0 10000 20000 30000 40000 50000 60000 70000
- 3. 5
- 3. 0
- 2. 5
- 2. 0
- 1. 5
- 1. 0
- 0. 5
0. 0
LnFl ow
4 5 6 7 8 9 10 11 12
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Regression Model 1• Log P-Conc = b0 + b1(Log Flow) + error
• Most simple model
• Historical use
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Regression Model 2• Log P-Conc = b0 + b1(Log Flow) + b2(Season) + error
• Addition of second independent variable “Season”
• Dual Slope Analysis
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Purpose of adding “Season”
- 3. 0
- 2. 5
- 2. 0
- 1. 5
- 1. 0
- 0. 5
0. 0
LnFl ow
5. 0 5. 5 6. 0 6. 5 7. 0 7. 5 8. 0 8. 5 9. 0 9. 5 10. 0 10. 5 11. 0
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Regression Model 3• Log P-Conc = b0 + b1(Log Flow) + b2(Season) +
b3(Season Effect) + error
• Addition of “Season Effect”• Interaction variable
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Purpose of adding “Season Effect”
• Interaction b/w two independent variables
• Slope adjustment
• Change in log TP concentration per unit flow during the winter season
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Results of Regression Models for the Maumee, WY 2005
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Selecting the Best Model for WY 2005
• Model 1 Results Intercept Log Flow Overall Model Mean Square
Estimate Estimate R² Significance Error
-3.1743 0.173 0.3091 0.1059
P-Value <.0001 <.0001 <.0001
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Selecting the Best Model for WY 2005
• Model 2 Results
Intercept Log Flow Season Overall Model Mean Square
Estimate Estimate Estimate R² Significance Error
-3.3331 0.2004 -0.1124 0.3218 0.1043
P-Value <.0001 <.0001 0.0167 <.0001
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Selecting the Best Model for WY 2005
• Model 3 Results
Intercept Log Flow Season Seas. Effect
Estimate Estimate Estimate Estimate R² Mod. Sig MSE
-2.2586 0.0451 -2.666 0.3297 0.4956 0.0778
P-Vals. <.0001 0.0405 <.0001 <.0001 <.0001
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Results of Regression Model 3 for the Maumee, WY 2003-2004
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Model 3: Viable Option?• Looked like a good choice for WY 2005
• Ran with WY 2003-2004 data
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Water Intercept Log Flow Season Season Effect Mod.
Year Estimate Estimate Estimate Estimate R² Sig.
2003 -3.9067 0.2893 -0.0442 0.0482 0.6061
P-values <.0001 <.0001 0.0462 0.0856 <.0001
2004 -3.511 0.2549 -1.8283 0.1745 0.6454
P-values <.0001 <.0001 <.0001 <.0001 <.0001
2005 -2.2586 0.0451 -2.666 0.3297 0.4956
P-values <.0001 0.0405 <.0001 <.0001 <.0001
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Estimating an Annual TP Load Using Regression Results
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Estimating an Annual Load With Regression
• Used Model 3
• Need to bring the log TP concentrations out of log-space (back-transforming)
• Back-transforming bias and estimated concentrations
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Bias Correction• To make up for the low bias….
• Total Phosphorus Concentration (ppm) =
Exp[LogPredicted P Concentration + (Mean Square Error * .5)]
• Estimating annual TP load from both measured and estimated data
• Couple conversion factors……Annual Estimated Load in metric tons/year
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What did We Find???
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Major Purpose of Our Research• The main objective- developing a daily
time series for accurately estimating an annual load for the Maumee in 2005
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How did the Regression Estimates Compare to the Beale Estimate?
• 95% Confidence IntervalsWater Regression Estimate Beale Estimate 95% Confidence
Year (Metric Ton/Year) (Metric Ton/Year) Interval
2003 2348.461 2341.401 2260.046 - 2422.757
2004 1905.47 1925.267 1829.385 - 2021.149
2005 2029.856 3134.59 2911.204 - 3357.975
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The Discrepancy
LnP_ Concent r at i on = - 2. 2585 +0. 0451 LnFl ow - 2. 666 Season +0. 3297 Season_ eff ect
N 313
Rsq 0. 4956
Adj Rsq0. 4907
RMSE 0. 2789
- 1. 00
- 0. 75
- 0. 50
- 0. 25
0. 00
0. 25
0. 50
0. 75
1. 00
Nor mal Quant i l e
- 3 - 2 - 1 0 1 2 3
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Problem with Regression
• Under-prediction
• Low-flow bias
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Future Directions• Improving the regression model
• Other independent variables
• More years
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Thank You
Any Questions?