First, I want to comment regarding Roger's
observation of nonlinear variables being acceptable in
regression analysis.
Yes, I like to think in terms of the
requirement being as if the equation is a function of the
coefficients, say beta_0, beta_1, beta_2, .... , beta_i, ...
, beta_N, where N+1 is the number of columns in the array of
independent variable values (the first column is all 1's). I
find it acceptable to think this way because during the
regression model development stage it is the values of the
beta's which are actually unknowns to be found in the
regression "least squares" process; only once the beta's are
estimated by this procedure, are the x-values, treated as
variables in the prediction phase.
The downside of thinking this way is that the
criterion of "least squares (error)" is only perfectly true
if the independent variable values (x_i's) are really
linearly related. If not, the criterion is only approximate.
Second, one of my key research areas was showing how
in using a regression-derived prediction equation using
nonfuzzy independent variables, it can be important to use
fuzzy variable values for some of the independent variables
yielding fuzzy predicted values. A case for which this can
be relevant is when historical data which include behaviors
of competitors are used to develop the regression model.
During the model development stage, the data are known
because they are historical, and nothing need be fuzzy. But
when the model is later used for prediction, competitors'
behavior is no longer known, so a fuzzy variable may
describe competitors' behaviors better. The research
challenge is in reflecting and interpretting the two aspects
of uncertainty: statistical margin of error and fuzziness
deriving from incomplete knowledge.
(B=)
On Sat, 19 Aug 2006, Roger Hui wrote:
+ Although %. is described as doing linear regression,
+ it can be used in cases where y is not a linear
+ function of x . e.g.:
+
+ x=: 10 [EMAIL PROTECTED] 100
+ y1=: (17 * 1.3 ^ x) + 10 [EMAIL PROTECTED] 0
+ ] c=: (^.y1) %. 1 ,. x
+ 2.83334 0.262363
+ ^ c
+ 17.0022 1.3
+ x ,. y1 ,. ^ (1 ,. x) +/ .* c
+ 27 20273.5 20274.8
+ 52 1.43055e7 1.43062e7
+ 25 11996.5 11997
+ 87 1.39162e11 1.39161e11
+ 67 7.3224e8 7.32257e8
+ 97 1.91847e12 1.91842e12
+ 11 304.737 304.701
+ 95 1.13519e12 1.13516e12
+ 85 8.23444e10 8.23439e10
+ 78 1.31229e10 1.3123e10
+
+
+ ----------------------------------------------------------------------
+ For information about J forums see http://www.jsoftware.com/forums.htm
+
(B=) <----------my "sig"
Brian Schott
Atlanta, GA, USA
schott DOT bee are eye eh en AT gee em ae eye el DOT com
http://schott.selfip.net/~brian/
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