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Multiple Linear Ratings

Multiple 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.

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Multiple Linear Ratings

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  1. Multiple 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. • Evaluate regression statistics and plots . If not acceptable determine if plots have non-linear trend. • If non-linear trend exists, begin steps for Multi-linear regression

  4. Multiple Linear Ratings Equation is of the form Y=aX1 + bX2 + nXn + C • Y = computed mean velocity • a,b,n = slope coefficients for each independent variable, X1, X2, Xn • C = y-intercept or “constant”

  5. Multiple Linear Ratings • Forms for sites where Vi and stage are significant: V = aVi + bVistage + C;or, V = Vi(a + b*stage) + C • The equations above are equivalent • Note - Vi * stage = second parameter. This yields similar results to using stage as the second parameter (can be implemented in ADAPS)

  6. Is Stage Related to Vmean?

  7. Sites with Index Rating that has Stage as a Variable • Large range in stage, relative to total depth • Large curvature in vertical velocity profile (rough channel bed) • Curvilinear velocity-velocity rating • Low R2 in velocity-velocity rating

  8. 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

  9. Multiple Linear Ratings

  10. Multi-Linear Regressions in Excel In Regression window, select Y data (Vmean) and X data (Vi) and X2 (Vi*stage) 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.

  11. Multi-Linear Regressions in Excel • Adjusted R2 (Don’t use R2) • Standard Error of the Estimate • # of Observations • P-value • Residual plots, (one for each independent variable) • Plot of computed versus measured-mean velocity

  12. Multi-Linear Regressions in Excel • Adjusted R2 (Don’t use R2) • Standard Error of the Estimate • Number of observations • P-value • Residual plots, (one for each independent variable) • Plot of computed versus measured-mean velocity

  13. Multi-Linear Regressions in Excel Examine residual plot for each variable for patterns

  14. Multi-Linear Regressions in Excel

  15. Let’s go through an example…

  16. Example • 22 Qms • Vmean: 0.25 – 1.15 ft/s • Vi: 0.44 – 1.25 ft/s • Stage: 0.2 – 3.4 ft • Vy: -0.10- 0.10 ft/s Look For Linear Relationships

  17. Example, cont.. Analysis results from Simple Regression

  18. Example, cont.. Simple Regression Results • Examine residual plots • Any patterns?

  19. Rating • Use coefficients to create the rating equation for your Simple Regression: Vmean = 0.99 * Vi – 0.17

  20. Example, cont.. Analysis results from Multi-Linear Regression Vmean = Vi*.82+ (Vi*Stage) * -0.15 + 0.16

  21. Example, cont. Examine Residuals

  22. Example, cont. Plot and Compare

  23. Let’s do an exercise….

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