WSP08546 - Laserfiche WebLink

Home Browse Search

N o ..'... 0') term (a*) as a function of time. Mueller and Osen evaluated the intercept term using a single-harmonic function with a period of 1 year: 031, = 030 + 031 sine~) + 032 cose~) (4) where m = month of the year; and - - ao, ai, and a2 = regression coefficients. Combining equations 3 and 4 yields: In(C) = 030 + aJ: sin e~) + 032 cos e~) + b In(Q). Mueller and Os en (1987) used a model of this type to estimate dissolved- solids discharge at sites in the Upper Colorado River basin. They found that the residual variance from a simple least-squares fit was not usually con- stant, which would imply a violation of the constant-variance assumption. To achieve constant variance, Mueller and Osen used weighted regression. Mo~t of the inequality in residual variance was associated with variation in stream- flow, which was correlated with month of the year. Therefore, weighting factors were computed based on the inverse of the monthly residual variance from the simple least-squares fit. When the linearized model (eq. 5) is detransformed to give estimates of dissolved-solids concentration in original units, it becomes: C = exp[ao + 031 sine~)+ 032 cose~)Ji. Miller (1984) showed that detransformation of a calibrated model can produce a biased estimator of the mean response. Therefore, a bias-correction factor must be included in the detransformed equation. For a model calibrated to natural logarithms and with normally distributed residuals, Miller recommended the bias-correction factor (BC) as: BC = exp (>.&2) (7) where &2 is the mean square error, which is an estimator of the residual variance. Miller's formulation was based on ordinary least-squares regres- sion, and must be modified for use with weighted least-squares regression. In a weighted least-squares solution, the residual variance is assumed to be 9 '" .., ,~ .f, " "1 ;1 <'j .~ (5 ) (6)