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1. Exploratory Data Analysis
1.3. EDA Techniques
1.3.3. Graphical Techniques: Alphabetic
1.3.3.27. Spectral Plot

1.3.3.27.3.

Spectral Plot: Sinusoidal Model

Spectral Plot for Sinusoidal Model spectral plot for sinusoidal model
Conclusions We can make the following conclusions from the above plot.
  1. There is a single dominant peak at approximately 0.3.
  2. There is an underlying single-cycle sinusoidal model.
Discussion This spectral plot shows a single dominant frequency. This indicates that a single-cycle sinusoidal model might be appropriate.

If one were to naively assume that the data represented by the graph could be fit by the model

    \[ Y_{i} = A_0 + E_{i} \]
and then estimate the constant by the sample mean, the analysis would be incorrect because
  • the sample mean is biased;
  • the confidence interval for the mean, which is valid only for random data, is meaningless and too small.
On the other hand, the choice of the proper model
    \[ Y_{i} = C + \alpha\sin{(2\pi\omega t_{i} + \phi)} + E_{i} \]
where α is the amplitude, ω is the frequency (between 0 and .5 cycles per observation), and \( \phi \) is the phase, can be fit by non-linear least squares. The beam deflection data case study demonstrates fitting this type of model.
Recommended Next Steps The recommended next steps are to:
  1. Estimate the frequency from the spectral plot. This will be helpful as a starting value for the subsequent non-linear fitting. A complex demodulation phase plot can be used to fine tune the estimate of the frequency before performing the non-linear fit.

  2. Do a complex demodulation amplitude plot to obtain an initial estimate of the amplitude and to determine if a constant amplitude is justified.

  3. Carry out a non-linear fit of the model

      \[ Y_{i} = C + \alpha\sin{(2\pi\omega t_{i} + \phi)} + E_{i} \]
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