Digital Signal Processing Reference
In-Depth Information
The denominator of equation (14-27) is also known as the
standard error of
regression
(SE
i
) for coefficient
b
i
. So the
t
-statistic is the ratio of the fit coefficient
to the standard error for a given model term.
For this test we reject the null hypothesis for coefficient
b
i
if
|
t
0
i
|
>t
α/
2
,n
−
(k
+
1
)
(14-29)
where
t
α/
2
,n
−
k
−
1
is the critical value for the
t
-distribution at a confidence level
1
1 degrees of freedom.
In essence, we are testing whether or not the fit coefficient, the numerator
of (14-27), has a greater effect on the model response than does the error (the
denominator) introduced by a given model term. If it does, we reject the null
hypothesis and conclude that the term has a significant effect. If not, we conclude
that the term does not have a significant effect and can exclude it from the model.
Appendix D provides critical values for the
t
-distribution at various confidence
levels and degrees of freedom.
−
α
with
n
−
k
−
Case Study Application
Hypothesis testing involves calculating the individual
t
-statistic for 21 fit coefficients each for the eye height and eye width, the details
of which are omitted. (Clearly, this suggests the use of a computer tool to auto-
mate the process!) We have actually already calculated the values for
σ
2
in
Section 14.5.1. They are the mean-squared-error terms in Table 14-6, which have
values of 10.260 and 8.946 for the eye height and width, respectively. Table 14-8
summarizes the calculations for the individual
t
-statistics. The diagonal elements
of the covariance matrix
C
i,i
, listed in the second column of the table, are used
along with the error variance terms to calculate the standard error for each term.
The
t
-ratios are calculated from the estimated fit coefficients and standard errors
according to equation (14-27).
At a 95% confidence level, the critical value for the
t
-statistic is 2.365 obtained
from Appendix D. Results of the hypothesis tests are listed in the rightmost
columns of the table, which show that 11 of the 20 model terms are significant for
the eye height model, while seven terms are significant for the eye width model.
All other terms can be removed from the model without significant degradation
in the model fit (see Problems 14-1 and 14-2).
14.6 CONFIDENCE INTERVALS
Our model allows us to estimate the predicted response to a given set of input
conditions. In creating the model, we have assumed that the errors are random,
uncorrelated, and fit a normal distribution. It follows that any prediction
y
will
also be a random variable that has an associated probability distribution. In fact,
the value predicted is the mean of a probability distribution that is determined
by the experiment design, the mean-squared error, and the set of input values for
which we are predicting the response.
ˆ
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