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Regression Type Window
This window lets you choose which type of leat-squares or regression calculation to perform. Five forms of regression / least squares fit calculations are supported:
Linear regression
This method calculates a least-squares fit of a straight line to a set of data pairs (x, y).
Assuming that each value of y
is equal to A + B.x + error, the values of A and B are
calculated in a manner that causes the sum of the squares of the error values to be minimised.
The image shows a straight line fitted to some randomly chosen data.
Polynomial regression
This approach assumes that each value of y is precisely the value of the
polynomial A0 + A1.x + A2.x2 +
A3.x3 ... + error, and the values of the coefficients A0,
A1, A2, A3, etc are calculated in such a manner as to minimise the sum of the squares of the error values.
The image shows the same data with a quadratic fitted to it.
Multivariate regression
This assumes that each value of y is equal to the expression A0.X0 + A1.X1 + A2.X2 + ... + error,
and chooses values of A0, A1, A2, etc so as to minimise the sum of the squares of
the error values.
Because of the multi-dimensional nature of the problem, UltimaCalc does not show a plot in this case.
Absolute deviation
This regression is similar to linear regression in that it attempts to fit a straight line to the data, but it minimises the sum of the absolute values of the errors rather than the sum of their squares. This helps to reduce the 'distorting' effect of outlying values.
The image shows the same data once more, with an absolute deviation fit (the thicker line, in orange, seemingly passing through three of the data points). The usual linear regression line is also shown.
Nonlinear Least Squares Fit
This method attempts to fit an arbitrary expression to the data.
Again we have the same data, and have assumed (perhaps on theoretical grounds) that the true underlying
relationship is of the form y = a + b.(1-exp(-x/c))^2. The problem then is to
calculate the values of a, b and c which give the best fit.
Where appropriate, i.e. for polynomial regression, multivariate regression, and nonlinear least squares fit, you will need to enter the
required order, the number of variates, or the number of variables. The order of a polynomial is the greatest power of its (main) variable. You can also change these values from the main
regression window.
No Choice Needed?
If you intend to load a previously saved regression data file, the initial choice of regression type is irrelevant, as it will be set automatically to match the data when it is loaded. Also, bear in mind that you can generally move from one mode to another while keeping the same data.