GG 313 Geological Data Analysis # 18 On Kilo Moana at sea October 25, 2005 Orthogonal Regression: Major axis and RMA Regression

Homework due - Any problems? Is the fit meaningful?

Curve fitting: The formula for obtaining each of the curve fitting algorithms is the same: 1)Decide what constitutes an “error”. 2)Decide what kind of curve parameters will be matched. 3)Take the first derivative of the error function with respect to each unkown. 4)Solve the resulting simultaneous equations for the unknowns where the derivative (slope) = 0.

Major Axis: In this case of orthogonal regression, the uncertainty is assumed to be in both x and y equally, and the best-fit curve minimizes the distance of each observation from the best fit line:

We will minimize the sum of the squares of the perpendicular distances from each point to a line: (4.13) The upper case letters are the observations and the lower case are the coordinates of our best-fit line. The line is defined by: (4.14) Or: (4.15)

The problem is to minimize the errors, but with the two constraints provided by minimizing E and fitting the line, we need help. Such help is provided by a method called Lagrange multipliers, where we form a new function, F, by adding the original error function(4.13) and the constraint equations (4.15). Each constraint is scaled by an unknown constant, I : (4.16) We now have an equation we can differentiate, finding the necessary partial derivatives: (4.17)

Looking at each partial derivative in turn: (4.18) (4.19) (4.20) (4.21) Each i represents a different equation, thus, from 4.18 and 4.19: (4.22) (4.23)

Now we go bak to (4.14) and plug in x i and y i from the equations above, and solve for i : (4.25) This gives us n+2 equations and n+2 unknowns (n i ’s and a and b). Substituting i from 4.25 into 4.20 and 4.21 yields: (4.27) (4.26) (4.28)

From (4.26) we see that (4.29) (4.30) After some algebra, and letting and noting that we finally solve for the slope: (4.32)

This gives us two solutions for b that are orthogonal to each other. One is the actual slope and the other is perpendicular to it. Reduced Major Axis (RMA) Regression:

In RMA regression we minimize the areas of the rectangle formed by the data points and the nearest point on the regression line. (4.33) The constraint is still a straight line, y i =a+bx i, and the Lagrange multiplier method leads to the equations like 4.16 and 4.17 giving: (4.37) (4.35) (4.36) (4.34)

Since we get again (4.30): From 4.34 and 4.35, we get: Substituting into the equation for a line, (4.38) (4.39) (4.41) (4.40) (4.42)

After plugging into (4.37), and more algebra, we get: (4.43) Thus, the Reduced Major Axis regression line is particularly simple to calculate, requiring only the means and standard deviations of the X i and Y i.