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Regression Analysis and Curve Fitting

Regression Methods

Aabel provides methods for different regression types, including:

Linear (X on Y)
Linear (thru zero)
Major axis
Reduced major axis
Polynomial
Exponential
Logarithmic
Power
Cubic spline interpolation
User-defined, non-linear regression
Multiple regression
Partial least squares regression (PLS)
Logistic regression

Regressions Concerning Two Continuous Variables

The regression methods grouped under this title either deal with finding the relationship between one outcome (dependent) variable and one predictor (independent) variable, or finding the relationship between two variables where the designation of dependent and independent variables is irrelevant. You can choose between 8 different methods (see the first 8 methods outlined above).

Performing curve fitting on an existing chart displays the fitted curve and the regression equation(s).
Performing regression analysis from the Stats Analyzer provides the options of (a) displaying the X-Y plot and the defined regression curve together with the regression parameters and an ANOVA report (if applicable) in a table format displayed on the viewer page, and (b) storing the results of the analysis (i.e., the observed and predicted sequences) in an auto-generated worksheet.

Cubic Spline Interpolation

A cubic spline is made from piece-wise third-order polynomials that pass through the control points provided.

You can apply a cubic spline interpolation to a displayed chart, or generate a specified sequence of interpolations (using the Stats Analyzer) and save them in a worksheet for other uses.

User-Defined Non-Linear Regression

The user-defined regression uses Levenberg-Marquardt "full Newton-type" method.

Aabel is distributed with a library of functions for use with the user-defined regression analysis.
Any calculated new functions and their parameters can be stored as a library function for later use.
Aabel uses an interactive graphical interface for the user-defined regression analysis.

Multiple Regression

Multiple regression is concerned with finding an equation that relates a single dependent variable to two or more independent variables.

The predicted and residual values, the regression parameters, as well as the original data will be placed in an auto-generated worksheet.
The ANOVA report for multiple regression, using k independent variables, displays the F statistic for testing the null hypothesis.
Additional output can include the partial regression ANOVA report for testing the significance of individual predictor variables.

Partial Least Squares Regression (PLS)

Partial least squares regression is an extension of the multiple linear regression model. In Aabel, you can predict one or more dependent variables from a set of independent (predictor variables).

PLS regression can be performed on a single data set (as for multiple regression), or using a training data set for predicting unknowns, new events, etc.
To use a calibration (training) data set, you can run a PLS regression on a representative data set and check the performance of the model before using it for predictive purposes.
The predictor coefficients and the predictions will be stored in two auto-generated worksheets.

Logistic Regression

Logistic regression allows you to predict a discrete outcome from a set of independent variables that may be continuous, discrete, or binary. The dependent variable is binary/dichotomous/binominal. Logistic regression in Aabel includes probability and logit (probability). Aabel allows generating:

Probability charts with one independent variable (see the right-hand side diagram below)
Probability charts with multiple dimension projections (see the left-hand side diagram below)
The observed probability can be displayed using markers, as shown in the images below. Probability charts with multiple dimension projections also generate a table of statistics (see the bottom image below).

The Logistic regression results displayed above were generated in Aabel using the published data of Pine, R.W. Wertz, M.J., Lennard, E.S., Dellinger, E.P., Carrico, C.J., and Minshew, H. (1983).