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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 Streamflow Record Computation using ADVMs and Index Velocity Methods Office of Surface Water
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
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”
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)
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
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
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.
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
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
Multi-Linear Regressions in Excel Examine residual plot for each variable for patterns
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
Example, cont.. Analysis results from Simple Regression
Example, cont.. Simple Regression Results • Examine residual plots • Any patterns?
Rating • Use coefficients to create the rating equation for your Simple Regression: Vmean = 0.99 * Vi – 0.17
Example, cont.. Analysis results from Multi-Linear Regression Vmean = Vi*.82+ (Vi*Stage) * -0.15 + 0.16
Example, cont. Examine Residuals
Example, cont. Plot and Compare