Information Technology Reference
In-Depth Information
In general, possible approximate statistical hypotheses are
•
In Germany the heights of the males are approximately normally distributed.
•
The Swedish males are slightly taller than the Italian males.
•
Most Swedish males are very likely fairly tall.
•
The people in the Nordic countries usually drink coffee quite much.
•
There is often fairly high positive correlation between the grades in mathematics
and physics among pupils at school.
•
One-dollar bills fairly likely contain a few particles of cocaine.
Within statistical explanations we may also use probabilistic or statistical state-
ments. For example, consider first the fact that the tossing of the coin in the real
world yields approximately 50 % of heads. If we now ask why approximately 50
% of these outcomes are heads, we can provide an approximate explanation that
it is due to this approximate frequency probability. Second, we may explain that
most Swedish men are tall because, according to the statistics, their average height
is approximately 180 cm.
In addition to statistical decision making, computational intelligence may provide
us with enhanced methods in model construction. We consider this aspect next.
18.3
Model Construction for Regression Analysis
Regression models are used when we aim at explaining or predicting the behavior
of a given variable, the dependent variable, according to the other variables known
as the independent variables. In the traditional case we presuppose that the indepen-
dent variables have linear correlations with the dependent variable, and thus linear
models are widely used. In practice we apply then the linear regression equation
=
∑
i
Y
a
i
X
i
+
b
,
i
=
1
,
2
,...,
n
.
(18.1)
in which
Y
is the dependent variable and
X
i
are the independent variables. When the
regression coefficients,
a
i
and
b
,
are specified correctly, this equation yields a plane
in a space with
n
1 dimensions according to the given data points [9],[28] .
Unfortunately, the linear models often seem to be too coarse for our purposes
because the real world is usually non-linear by nature. Thus, we may also attempt
to apply non-linear models, but then we do not necessarily know which regression
function would be appropriate to us, and we may have thousands of alternatives.
Even if we can find such function, it may be complicated and laborious with respect
to calculations, and naturally deep mathematical knowledge is also required.
Computational intelligence may resolve several of these problems, and we will
apply it with a synthetic data set of 98 persons which for the sake of simplicity only
comprises three variables, age in years, body mass index (
Bmi
) and systolic blood
pressure in mmHg (
Syst
). This data set is a modified version that is presented in
[10]. We apply SPSS
TM
+
and MATLAB
TM
software to our analyses below.
