Modeling with Neural Networks: Principles and Model Design Methodology - Neural Networks: Methodology and Applications - page 92

2.2.5 Confidence Interval

The estimation of a confidence interval provides an elegant solution to the
problem that has just been mentioned.
A confidence interval , with confidence threshold 1 − α , for a random vari-

able Y , is an interval that, with probability 1 − α , contains the value of the

expectation of Y .
Thus, instead of simply estimating the true value of the temperature by
averaging the results of measurements performed presumably under identical
conditions, one can estimate an interval within which the true value of the
temperature is to be found, with probability 1 α .Thisisamuchmore

useful and significant information: the smaller the confidence interval, the
more confident one can be in the estimate of the quantity of interest.
The procedure for computing a confidence interval, and an example, are
described in the additional material at the end of the chapter.

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2.2.6 Hypothesis Testing

Hypothesis testing is a conventional statistical technique that aims at estimat-
ing whether a given hypothesis about a model is significantly in agreement,
or in disagreement, with experimental data. In the field of modeling, the hy-
potheses that are tested are related to the model that is being designed.
A hypothesis, called “null hypothesis” H 0 , and its complement H 1 ,are
stated. A risk α of rejecting the null hypothesis H 0 although it is valid, is
chosen. Then the design of a hypothesis test consists in

• finding a random variable, whose distribution is known if the null hypoth-

esis is true, and a realization of which can be computed from the available
experimental data;

• computing that realization.

If the probability of the latter lying in a given interval is too low given the
distribution of the random variable, the null hypothesis has a low probability
of being true, hence it is rejected. An example of hypothesis testing is provided
in the additional material at the end of the chapter.

2.3 Static Black-Box Modeling

In the previous section, the basic elements of point estimation were explained:
a measurable quantity was considered, and modeled as a random variable, its
expectation value and variance were estimated, and a confidence interval was
computed, from the available measurements performed under identical condi-
tions. The process being the temperature of a fluid in an oven, it was assumed
that all measurements were performed with a given heater intensity, a given
external temperature, etc. Disturbances might be the intrinsic noise of the