1-1

Appendix 1. LOADEST Model Summaries for Rejected Models

Rejected LOADEST Models from Klamath River at Keno (site KRK)

Total Phosphorus

> print(TP.keno.sel, brief=F,load.only=T)

 LOADEST

 A Program to Estimate Constituent Loads

U.S. Geological Survey, Version for R 0.1 (June, 2013)

------------------------------------------------------

Station: Klamath River at Keno, OR

Constituent: TP

----------------------------------------------------------------------

 Constituent Output File Part Ia: Calibration (Load Regression)

----------------------------------------------------------------------

 Number of Observations: 75

Number of Uncensored Observations: 75

 Center of Decimal Time: 2013.73

 Center of ln(Q): 6.74

 Period of record: 2012-04-10 to 2015-03-23

Model Evaluation Criteria Based on AMLE Results

-----------------------------------------------

 model AIC SPCC AICc

1 1 104.68 111.64 105.02

2 2 102.76 112.03 103.33

3 3 106.51 115.78 107.08

4 4 48.21 59.80 49.08

5 5 103.68 115.27 104.55

6 6 48.13 62.04 49.37

7 7 49.44 63.35 50.68

8 8 49.96 66.18 51.63

9 9 48.61 67.15 50.79

Model # 6 selected

Selected Load Model:

--------------------

TP ~ model(6)

where:

 TP is the constituent load in log(kg/d)

and model 6 has these variables:

 lnQ is ln(Q) - center of ln(Q)

 lnQ2 is ln(Q) - center of ln(Q))^2

 sin.DECTIME is sine(2 * pi * decimal time)

 cos.DECTIME is cosine(2 * pi * decimal time)

Model coefficients:

 Estimate Std. Error z-score p-value

(Intercept) 5.61015 0.04465 125.651 0.0000

lnQ 0.76527 0.10181 7.517 0.0000

lnQ2 -0.15831 0.11300 -1.401 0.1499

sin.DECTIME 0.06205 0.05792 1.071 0.2695

cos.DECTIME -0.50409 0.05537 -9.103 0.0000
1-2

AMLE Regression Statistics

Residual variance: 0.1016

R-squared: 77.6 percent

G-squared: 112.2 on 4 degrees of freedom

P-value: <0.0001

Prob. Plot Corr. Coeff. (PPCC):

 r = 0.995

 p-value = 0.7404

Serial Correlation of Residuals: 0.563

Correlation Between Explanatory Variables

-----------------------------------------

 lnQ lnQ2 sin.DECTIME

lnQ2 0.0000 

sin.DECTIME 0.3645 0.1643 

cos.DECTIME -0.3316 0.1646 -0.0204 

Correlation Between Variable Coefficients

-----------------------------------------

 lnQ lnQ2 sin.DECTIME

lnQ2 0.0064 

sin.DECTIME -0.3754 -0.1600 

cos.DECTIME 0.3428 -0.1578 -0.0862 

Variance Inflation Factors:

 VIF

lnQ 1.312

lnQ2 1.058

sin.DECTIME 1.199

cos.DECTIME 1.167

Comparison of Observed and Estimated Loads

------------------------------------------

 The summary statistics and bias diagnostics presented below are based

on a comparison of observed and estimated loads for all dates/times within

the calibration data set. Although this comparison does not directly

address errors in load estimation for unsampled dates/times, large

discrepancies between observed and estimated loads are indicative of a

poor model fit. Additional details and warnings are provided below.

Note: The comparison that follows uses a concentration equal to 1/2 the

detection limit when an observation is censored. The summary stats and

bias diagnostics are therefore slightly inaccurate for censored datasets.

 Summary Stats: Loads in kg/d

---------------------------------------------

 Min 25% 50% 75% 90% 95% Max

Est 53.9 157 278 381 499 638 751

Obs 48.0 147 244 445 542 704 856

Bias Diagnostics

----------------

Bp: -0.2487 percent

PLR: 0.9975

 E: 0.6406

where:

 Bp Load Bias in Percent

 Positive (negative) values indicate over (under) estimation.

 ***The model should not be used when the + or - bias exceeds 25%***

 PLR Partial Load Ratio

 Sum of estimated loads divided by sum of observed loads.

 Values greater than 1 indicate over estimation.
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 Values less than 1 indicate under estimation.

 E Nash Sutcliffe Efficiency Index

 E ranges from -infinity to 1.0

 E = 1; a perfect fit to observed data.

 E = 0; model estimates are as accurate as the mean of observed data.

 E < 0; the observed mean is a better estimate than the model estimates.

NOTE: Additional information on model calibration is included in the

 residual diagnostic plots. users should conduct a thorough

 residuals analysis. Example residual plots are shown in figures

 7, 8, 9, and 17 of the LOADEST documentation (Runkel et al., 2004).

> print(TP.keno.4, brief=F,load.only=T)

 LOADEST

 A Program to Estimate Constituent Loads

U.S. Geological Survey, Version for R 0.1 (June, 2013)

------------------------------------------------------

Station: Klamath River at Keno, OR

Constituent: TP

----------------------------------------------------------------------

 Constituent Output File Part Ia: Calibration (Load Regression)

----------------------------------------------------------------------

 Number of Observations: 75

Number of Uncensored Observations: 75

 Center of Decimal Time: 2013.73

 Center of ln(Q): 6.74

 Period of record: 2012-04-10 to 2015-03-23

Model Evaluation Criteria Based on AMLE Results

-----------------------------------------------

 model AIC SPCC

1 4 48.21 59.8

Model # 4 selected

Selected Load Model:

--------------------

TP ~ model(4)

where:

 TP is the constituent load in log(kg/d)

and model 4 has these variables:

 lnQ is ln(Q) - center of ln(Q)

 sin.DECTIME is sine(2 * pi * decimal time)

 cos.DECTIME is cosine(2 * pi * decimal time)

Model coefficients:

 Estimate Std. Error z-score p-value

(Intercept) 5.58007 0.03941 141.5916 0.0000

lnQ 0.76618 0.10250 7.4751 0.0000

sin.DECTIME 0.04906 0.05756 0.8524 0.3822

cos.DECTIME -0.51633 0.05505 -9.3793 0.0000

AMLE Regression Statistics

Residual variance: 0.1029

R-squared: 76.97 percent

G-squared: 110.1 on 3 degrees of freedom

P-value: <0.0001

Prob. Plot Corr. Coeff. (PPCC):

 r = 0.9958

 p-value = 0.8354
1-4

Serial Correlation of Residuals: 0.5546

Correlation Between Explanatory Variables

-----------------------------------------

 lnQ sin.DECTIME

sin.DECTIME 0.3645 

cos.DECTIME -0.3316 -0.0204 

Correlation Between Variable Coefficients

-----------------------------------------

 lnQ sin.DECTIME

sin.DECTIME -0.3793 

cos.DECTIME 0.3482 -0.1144 

Variance Inflation Factors:

 VIF

lnQ 1.312

sin.DECTIME 1.169

cos.DECTIME 1.138

Comparison of Observed and Estimated Loads

------------------------------------------

 The summary statistics and bias diagnostics presented below are based

on a comparison of observed and estimated loads for all dates/times within

the calibration data set. Although this comparison does not directly

address errors in load estimation for unsampled dates/times, large

discrepancies between observed and estimated loads are indicative of a

poor model fit. Additional details and warnings are provided below.

Note: The comparison that follows uses a concentration equal to 1/2 the

detection limit when an observation is censored. The summary stats and

bias diagnostics are therefore slightly inaccurate for censored datasets.

 Summary Stats: Loads in kg/d

---------------------------------------------

 Min 25% 50% 75% 90% 95% Max

Est 65.9 152 279 377 490 689 866

Obs 48.0 147 244 445 542 704 856

Bias Diagnostics

----------------

Bp: 0.4955 percent

PLR: 1.005

 E: 0.6402

where:

 Bp Load Bias in Percent

 Positive (negative) values indicate over (under) estimation.

 ***The model should not be used when the + or - bias exceeds 25%***

 PLR Partial Load Ratio

 Sum of estimated loads divided by sum of observed loads.

 Values greater than 1 indicate over estimation.

 Values less than 1 indicate under estimation.

 E Nash Sutcliffe Efficiency Index

 E ranges from -infinity to 1.0

 E = 1; a perfect fit to observed data.

 E = 0; model estimates are as accurate as the mean of observed data.

 E < 0; the observed mean is a better estimate than the model estimates.

NOTE: Additional information on model calibration is included in the

 residual diagnostic plots. users should conduct a thorough

 residuals analysis. Example residual plots are shown in figures

 7, 8, 9, and 17 of the LOADEST documentation (Runkel et al., 2004).
1-5

Total Nitrogen
1-6
1-7

Five-Day Biochemical Oxygen Demand (BOD5)
1-8

Five-Day Biochemical Oxygen Demand (BOD5)
1-9
1-10

Five-Day Carbonaceous Biochemical Oxygen Demand (CBOD 5)
1-11
1-12

Rejected LOADEST Models from Upper Klamath Lake at Fremont Bridge (site FMT)

Total Phosphorus
1-13
1-14

Total Nitrogen
1-15
1-16
1-17
1-18

Five-Day Biochemical Oxygen Demand BOD5
1-19
1-20
1-21

Five-Day Carbonaceous Biochemical Oxygen Demand (CBOD5)
1-22
1-23