Digital Signal Processing Reference
In-Depth Information
As a result, we must account for the error sources when using the model to
make decisions based on the responses predicted. The way in which we do so
is to determine a confidence interval around the response, which is estimated
from the
t
- distribution at a specified confidence level (1
−
α
) and the number of
degrees of freedom associated with the error. In simple terms, a 95% confidence
interval accounts for 95% of the statistical variation in the prediction. Intuitively,
requiring increased confidence that a prediction falls within the calculated confi-
dence interval means that the width of the interval must also increase, so that it
accounts for more of the statistical variability.
Since the confidence interval depends on the input values, we start by con-
structing the input vector from the independent variables using the method out-
lined in Section 14.3:
1
,x
1
,x
2
,
...
,x
k
|
x
in
=|
(14-30)
The predicted value
y
is
y
=
x
in
·
b
(14-31)
We calculate the confident interval CI
y
from the
t
-distribution, estimated error
variance, input vector, and the covariance matrix:
CI
y
=
t
α/
2
,n
−
k
−
1
x
in
(
X
T
X
)
−
1
x
in
]
σ
2
[1
+
(14-32)
In equation (14-32), the term under the square root is the standard error of
prediction. At a given confidence level, the predicted response has a confidence
interval of
y
−
CI
≤
y
≤
y
+
CI
(14-33)
y
i
ˆ
y
i
ˆ
We can also calculate confidence intervals on the mean response for the experi-
mental observations using
CI
y
=
t
α/
2
,n
−
k
−
1
σ
2
[
x
in
(
X
T
X
)
−
1
x
in
]
(14-34)
The confidence interval around the response observed is smaller than those around
the responses predicted. Figure 14-2 includes confidence intervals calculated with
(14-34) for our example signaling system.
14.7 SENSITIVITY ANALYSIS AND DESIGN OPTIMIZATION
As stated in Section 14.1, the power of the response surface model is that it
provides a tool for understanding the behavior of a complicated system so that we
can adjust the design to create a working solution. This involves identifying the
factors that have the greatest influence on system performance and adjusting them,
within the limits of the manufacturing capability, to maximize the robustness of
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