If the quadratic term is negligible-- meaning that the linear term is sufficiently accurate without adding the quadratic term-- then linear interpolation is sufficiently accurate . ( If the problem is sufficiently important, or if the quadratic term is nearly big enough to matter, then one might want to determine whether the _ sum _ of the quadratic and cubic terms is large enough to matter in the problem .)
42.
If the quadratic term is negligible-- meaning that the linear term is sufficiently accurate without adding the quadratic term-- then linear interpolation is sufficiently accurate . ( If the problem is sufficiently important, or if the quadratic term is nearly big enough to matter, then one might want to determine whether the _ sum _ of the quadratic and cubic terms is large enough to matter in the problem .)
43.
I place the first two charges on the x-axis one unit-length away from the origin in oposit directions and the third at a small distance \ delta \, from the origin along a direction that makes an angle \ theta \, with the x-axis ( I don't like using roman letters for angles ) . \ delta \, is small enough that any quadratic terms ( or higher ) will be considered negligible and will be dropped.
44.
This observation has been used in extending neural network models; the " circular " case, which corresponds to introducing only the sum of pure quadratic terms \; x _ 1 ^ 2 + x _ 2 ^ 2 + x _ 3 ^ 2 \; \ ldots \; with no mixed products ( \; x _ 1 x _ 2, \; x _ 1 x _ 3 \; \ ldots \; ), has been proven to be the optimal compromise between extending the classifier's representation power and controlling the risk of overfitting ( Vapnik-Chervonenkis dimension ).
45.
If you want to test higher-order non-linearities, you can include x ^ 3, x ^ 4, etc . This is a better approach than comparing the goodness of fit for a couple of reasons : ( 1 ) without doing additional math, you don't know whether the improvement in the goodness of fit is statistically significant, ( 2 ) R ^ 2 is guaranteed to increase when you add the quadratic term, so you must use an adjusted R ^ 2-- but, there are several ( or none ) to choose from depending on what you intend to do with the data and whether your dependent variable is discrete or continuous . talk ) 16 : 17, 26 June 2010 ( UTC)
