1.
Exploratory Data Analysis
1.4.
EDA Case Studies
1.4.2.
Case Studies
1.4.2.5.
Beam Deflections
1.4.2.5.4.
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Validate New Model
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4-Plot of Residuals
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The first step in
evaluating the fit
is to generate a 4-plot
of the residuals.
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Interpretation
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The assumptions are addressed by the graphics shown above:
- The run sequence plot
(upper left) indicates that the data
do not have any significant shifts in location. There does
seem to be some shifts in scale. A start-up effect was
detected previously by the complex demodulation amplitude
plot. There does appear to be a few outliers.
- The lag plot
(upper right) shows that the data are random.
The outliers also appear in the lag plot.
- The histogram
(lower left) and the
normal probability plot
(lower right) do not show any serious non-normality in
the residuals. However, the bend in the left portion of the
normal probability plot shows some cause for concern.
The 4-plot indicates that this fit is reasonably good. However,
we will attempt to improve the fit by removing the outliers.
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Fit Results with Outliers Removed
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The following parameter estimates were obtained after removing three
outliers.
Coefficient Estimate Stan. Error t-Value
C -178.788 10.57 -16.91
AMP -361.759 25.45 -14.22
FREQ 0.302597 0.1457E-03 2077.00
PHASE 1.46533 0.4715E-01 31.08
Residual Standard Deviation = 148.3398
Residual Degrees of Freedom = 193
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New Fit to Edited Data
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The original fit, with a residual standard deviation of
155.84, was:
\( \hat{Y}_i = -178.786 - 361.766[2\pi(0.302596)T_i + 1.46536] \)
The new fit, with a residual standard deviation of 148.34, is:
\( \hat{Y}_i = -178.788 - 361.759[2\pi(0.302597)T_i + 1.46533] \)
There is minimal change in the parameter estimates and about a 5 %
reduction in the residual standard deviation. In this case, removing
the residuals has a modest benefit in terms of reducing the
variability of the model.
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4-Plot for New Fit
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This plot shows that the underlying assumptions are satisfied
and therefore the new fit is a good descriptor of the data.
In this case, it is a judgment call whether to use the fit
with or without the outliers removed.
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