This leads to solving a nonlinear system of 11 equations in the least-squares method
. The solution of this system is not given here due to its bulkiness.
It should be noted that, all the while in all of the aforementioned cases, the Eadie-Hofstee method has coefficients of variation that are highly comparable to those based on ordinary least-squares method
; the EH estimated [K.sub.m] and [V.sub.max] values are much further away from the true values than the estimates obtained by Hanes-Woolf, ordinary least-squares, and robust nonlinear regression methods.
It is for this reason that robust alternatives to the usual least-squares method
have not been properly used in practice (Efron, 1982).
We computed the estimators of the parametric components [[beta].sub.1] and [[beta].sub.2] and the nonparametric component [theta](*) by using the proposed integral least-squares method
. The results for the parametric components are reported in Table 2, and the results for the nonparametric components are reported in Figure 3, where the solid curve is the estimator of [theta](*) and the dashed curve is the pointwise confidence interval of [theta](*).
With the least-squares method
, (22) can be rewritten as
A similar algorithm based on the least-squares method
was described by Ubi (1989).
First, the accessible published motor efficiency and power factors are chosen as inputs; second, the nonlinear equivalent circuit equations are linearized by introducing three iterative ratios; then the linear least-squares method
is employed to obtain the circuit parameters; finally the motor efficiency is estimated by the equivalent circuit method with calculated circuit parameters.
The lower and upper limits of the 90 percent prediction band by the least-squares method
are parallel to the conditional mean curve, which is valid only if the variability in the data had remained constant across patient contact time.
The least-squares method
indicates the best function y(x) for our set of data.
Once ?i is estimated by applying the least-squares method
or IV method, the model parameters can be recovered.
Three frequently referenced spatial models (spherical, exponential, Gaussian) (Cressie 1993a; Davis 2002) were considered using the weighted least-squares method
(Gribov et al.
Linear regression is a special case of the least-squares method
and it is the most common way of analyzing strength data.