Least squares line fitting:                      by:  Edward T Peltzer, MBARI
                                                 revised:  2016 Mar 17.


Model-1 regressions:


        There are four m-files to use in this situation:

                lsqfitx.m  --  X on Y regression  (alternate or reversed regression)
                lsqfity.m  --  Y on X regression  (standard linear regression)
                lsqfityw.m  --  weighted-Y on X regression
                lsqfityz.m  --  weighted-Y on X regression (revised)


lsqfitx.m  --   X-on-Y regression.

                ALTERNATE or REVERSED linear regression equation.

                Fit line by minimizing x-residuals only; y-residuals are
                assumed to be zero or very small.  All data points are given
                equal weight.


lsqfity.m  --   Y-on-X regression.

                STANDARD or COMMON linear regression equation.

                Fit line by minimizing y-residuals only; x-residuals are
                assumed to be zero or very small.  All data points are given
                equal weight.


lsqfityw.m  --  WEIGHTED regression of a simple linear equation.

                [wY-on-X regression.]

                Fit line by minimizing y-residuals only; x-residuals are
                assumed to be zero or very small.  Data points are given
                varying weight.  Uncertainties for slope and intercept
                seem to be very small compared to those from lsqfity.m.

                If the same uncertainty is used for all the Y-data points,
                then the slope and intercept are the same as for lsqfity.m,
                but their calculated uncertainties are different.


lsqfityz.m  --  WEIGHTED regression of a simple linear equation.

                [wY-on-X regression.]

                This is the same regression as lsqfityw.m EXCEPT York's (1966)
                equations for the uncertainty in slope and intercept are used.
                These uncertainties are comparable to those from lsqfity.m.




Model II regressions:


        There are four m-files to use in this situation:

                lsqbisec.m  --  determination of the least squares bisector
                lsqcubic.m  --  weighted determination of the major axis
                lsqfitgm.m  --  determination of geometric mean
                lsqfitma.m  --  determination of major axis


lsqbisec.m  --  LEAST SQUARES BISECTOR.

                Slope of line is determined by bisecting the minor angle
                between the two model I regressions: Y-on-X and X-on-Y.
                The y-intercept is obtained by running the line through the
                centroid.  All data are given equal weight.

                Use when units or range of X and Y are different.

                This algorithm uses lsqfity.m and lsqfitx.m.  Make sure that
                the directory containing both of these files is listed in
                your MATLABPATH.


lsqcubic.m  --  LEAST SQUARES CUBIC -- Correlation of wX & wY.

                Weighted correlation of two independent variables.

                Line is fit by minimizing both x- and y-residuals simultane-
                ously for WEIGHTED data points.  Each data point is given its
                own weight either as the inverse-square of the measurement
                precision or as the inverse-square of the relative measurement
                precision for the method times concentration.

                This algorithm uses lsqfitma.m for the first estimate of slope.
                Make sure that the directory containing this file is listed in
                your MATLABPATH.  Iteration proceeds until the change in slope
                is less than the user defined limit.


lsqfitgm.m  --  GEOMETRIC MEAN REGRESSION, aka: the reduced major axis.

                Slope of line is the geometric mean of the two slopes
                determined by regressing Y-on-X and X-on-Y.  The y-intercept
                is obtained by running the line through the centroid.
                All data are given equal weight.

                Use when units or range of X and Y are different.

                This algorithm uses lsqfity.m and lsqfitx.m.  Make sure that
                the directory containing both of these files is listed in
                your MATLABPATH.


lsqfitma.m  --  MAJOR AXIS, aka: first principal component.

                Line is fit by minimizing BOTH x- and y-residuals simultane-
                ously.  All data are given equal weight.

                Use when units and range of X and Y are the same.




        For more info regarding Model I and II regressions and correlation
        analysis, see:

                BIOMETRY, 3rd edition  by  Sokal and Rohlf (1995).
                  W. H. Freeman and Company, San Francisco, CA.

                MATHEMATICAL METHODS FOR OCEANOGRAPHERS  by  E. A. Laws (1997).
                  John wiley & Sons, Inc., New York, NY.