Nonlinear Least Squares (NLLS) Regression allows the estimation of
paramters in user-defined equation. Draco uses a Generalized Least
Squares method to estimate parameters, as described in the Method section.
Usage
The NLLS regression can be started by clicking Nonlinear
Least Squares... in the Data
Window's Regression menu. The NLLS regression dialog appears
below:
The NLLS window has two basic panes, Model and Parameters.
The Model
window, shown above, is used to specify the equation to attempt to fit
to the data. The Combo Box on the far left specifies the
dependent variable for the regression. The equation to fit is
entered in the center box. Combo Boxes on the right are
present
for convenience to add existing variables and constants to the
equation. In the example above, alpha is a constant
in the worksheet, prob
is a variable from the worksheet, and b1 is a parameter
to be fit. This parameter is specified in the Parameters
tab.
The Parameters
tab lists all the parameters in the equation that will be estimated
along with their initial estimates. Paramters are added and
removed from this pane using the toolbar buttons:
The Plus sign adds a new parameter, and the Subtract sign removes the
selected parameter.
Draco
uses an iterative technique to arrive at the final estimates for the
Non-Linear Least Squares regression. The convergence criteria
can
be set in by clicking theIterations and
Convergence... item in the Options Menu; the
default values are usually sufficient.
Once all parameters are specified along with the equation, the
regression can be run by selecting Compute
Regression... from the Preform Fit Menu.
While running, the iterative progress dialog will appear, showing
progress of the regression.
Output
The Nonlinear Least Square Regression will generate output similar to the following:
Results of the Non-Linear Least Squares Regression Model
Regression Variable:
ser1
Sum Squared of the Residuals:
1.88226
Standard Error of the Fit:
0.26403
R-Squared Value:
1
Adjusted R-Squared Value:
1
Coefficient
Value
Std. Err.
t-Score
alpha
2.00019
.6331E-05
3.5508E04
beta
1.44938
0.07055
20.54421
Further information can be generated after the regression. By
selecting Generate Column Data from the Perform Fit
menu, the
estimated dependent variable data will be output as a new variable in
the Data window. The resulting covariance matrix can be
viewed by selecting View Covariance Matrix... from
the Supporting
Data menu. The residuals from the regression can also be
output to the Data window as a new variable by selecting Output
Residuals to Column from the Supporting Data menu.
Method
The Nonlinear Least Squares
Regression uses a Generalized Least Squares (GLS) technique to solve
the system. Each iteration solves the following equation:
minimize || B - A*X ||
In this equation, however, the A matrix is constructed from the
derivatives of the equation of interest with respect to the parameters
for estimation. In other words, the A matrix is constructed as:
The
matrix above actually collapses into the expected Least Squares form
for linear equations. The derivatives in Draco are evaluated
numerically by applying a step of 1*10-10 to the previous
iteration's (or the initial) estimates. The procedure is repeated
until the sum of all changes is within the absolute tolerance or the maximum number of iterations are completed. The solution to the equation is computed in eachiteration using the same LAPACK calls as outlined in the Least Squares Fit.
Please note that the method described is prone to divergence. An improved technique is planned for future versions.
