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Simple Linear Ratings

Simple Linear Ratings. Streamflow Record Computation using ADVMs and Index-Velocity Methods Office of Surface Water. Rating Development. Rating Development – Always Start with a Simple Linear Rating!. Simple Linear Ratings. Represented by a straight line: V mean = mV i + c

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Simple Linear Ratings

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  1. Simple Linear Ratings Streamflow Record Computation using ADVMs and Index-Velocity Methods Office of Surface Water

  2. Rating Development

  3. Rating Development – Always Start with a Simple Linear Rating!

  4. Simple Linear Ratings • Represented by a straight line: Vmean= mVi+ c WhereVmean= computed mean velocity Vi = index velocity m = line slope or “coefficient” c = y-intercept or “constant” • Index velocity is the only explanatory variable • Will use Ordinary Least Squares (OLS) regression techniques

  5. Linear Regression Assumptions • Dependent (e.g., Vmean) and independent variables (e.g., Vi) are linearly related • Independent variable is representative of dependent variable • Residuals have equal variance (random pattern) • Observed values of Vmean are uncorrelated, random events • Residuals are normally distributed • Independent variables can be measured with reasonable error

  6. + + Simple Linear Ratings Vmean = mVi + c Unidirectional flow Bi-directional flow

  7. Simple Linear Ratings Vmean = mVi+ c Graphical Representation Vmean rise run m = rise/run c Vi

  8. Vmean Simple Linear Ratings Slope Coefficient, m m = DVmean/ DVi m = 1 Assume c = 0 m=1 if Vmean= Vi (we are measuring mean channel velocity) m<1 if Vi> Vmean(measuring in a higher-velocity zone) m>1 if Vmean>Vi (measuring in a lower-velocity zone) m > 1 Vmean Vi m < 1 1 1 Vi

  9. Simple Linear Ratings c > 0 c = constant (aka offset or intercept) c = 0 c < 0 Vmean Vi

  10. Simple Linear Ratings When c = 0: Vmean =mVi + c and becomes Vmean = mVi, a simple coefficient rating Could be possible if: • The instrument measured a significant portion of the channel all of the time • The instrument measured the location of mean velocity • The range of stage and velocities were relatively small

  11. Simple Linear Ratings • For most sites, c will not equal zero • c is usually a much smaller number than “m”, the slope coefficient • c could still be an important factor for defining slope – not considering c and “forcing the rating through zero” can be detrimental

  12. Simple Linear Ratings • Once you evaluate plots, select possible candidates as the “index” velocity • Run regression statistics • Evaluate regression statistics and residuals plots • Compare ratings – which is best?

  13. Linear Regressions in Excel 1. Go to Data Ribbon then select Data Analysis 2. In Data Analysis window, scroll down and select Regression. Click OK.

  14. Linear Regressions in Excel If you don’t see the Data Analysis option, you need to load the add-in Go to the File Ribbon, then select Options

  15. Linear Regressions in Excel Select Add-Ins Next to Manage – make sure Excel Add-Ins is displayed. Click Go. Check box next to Analysis ToolPak. Click OK.

  16. Back to Excel Regression Steps….

  17. Linear Regressions in Excel 3. In Regression window, select Y data (Vmean) and X data (Vi) 4. Check box next to Labels if you selected the column header (Vmean or Vi) and want the X/Y variable labeled in the regression output.

  18. Linear Regressions in Excel 5. Pick where you want data output (easiest to put in a new worksheet tab within your workbook) 6. Check boxes next to Residuals, Residuals Plots, and Line Fit Plots 7. Click OK

  19. Linear Regressions in Excel 8. Review regression statistics (important ones highlighted in yellow here – more guidance later in presentation)

  20. Linear Regressions in Excel 9. Examine residual plot for patterns (more guidance later in presentation)

  21. Interpreting Regression Statistics

  22. Regression Statistics • Can be overwhelming • What to focus on? • What do they mean?

  23. Regression Statistics

  24. Statistics: R-Square • Do not rely on R2 alone when evaluating a regression • Can be misleading • R2 between just 2 points is a perfect 1! • These plots have the same R2

  25. Interpreting Residual Plots

  26. Residuals: What Are They? Vmean= mVi+ c e4 e3 e2 Dependent variable - Vmean e1 m residual – departure of observation from regression line c Independent variable - Vi

  27. Residuals: What are They? • In a linear regression, the line is computed so that the residuals are minimized • “Best” fit of all the data residual

  28. Residual Plots

  29. Residual Plots

  30. Let’s go through an example…

  31. Example • 41 Qms • Good range in velocity • Some gaps • Vmean: 0.04 - 1.79 ft/s • Vi: 0.06 – 2.03 ft/s

  32. Example, cont. • In Excel, run Regression Analysis

  33. Example, cont. • Examine residual plots • Any patterns?

  34. Rating • Use coefficients to create the rating equation: Vmean = 0.90 * Vi – 0.04

  35. Which Regression is Best? • Example: Comparing multiple ratings (e.g., range-averaged Vx, multi-cell Vx, etc) • Look at scatter plots then compare regression statistics

  36. Let’s do an exercise….

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