our week at math camp abridged
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Our Week at Math Camp Abridged. Group 2 π = [Erin Groark, Sarah Lynn Joyner, Dario Varela, Sean Wilkoff]. Agenda. Harmonic Oscillator Model Parameter Estimates Standard Errors Confidence Intervals Model Fit Residual Analysis Beam Model Model Fit Analysis Comparison.

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our week at math camp abridged

Our Week at Math CampAbridged

Group 2π =

[Erin Groark, Sarah Lynn Joyner, Dario Varela, Sean Wilkoff]

agenda
Agenda
  • Harmonic Oscillator Model
    • Parameter Estimates
      • Standard Errors
      • Confidence Intervals
    • Model Fit
    • Residual Analysis
  • Beam Model
    • Model Fit
    • Analysis
  • Comparison
harmonic oscillator model parameter estimates
Harmonic Oscillator Model:Parameter Estimates
  • C = 0.80406
    • Standard error: 0.011153
    • Confidence interval:

(0.7818, 0.8263)

  • K = 1515.7
    • Standard error: 0.43407
    • Confidence interval:

(1514.8, 1516.6)

  • How good are these estimates?
harmonic oscillator model model fit zooms
Harmonic Oscillator Model:Model Fit Zooms

Beginning Middle

Area of greatest deviation

Area of smallest deviation

harmonic oscillator model model fit1
Harmonic Oscillator Model:Model Fit
  • Model appears to fit best at the beginning
    • Peaks are same size
    • Closer examination reveals that the fit is worst there
  • Large amount of noise—another frequency interferes strongly at first
harmonic oscillator model residual analysis
Harmonic Oscillator Model:Residual Analysis
  • Statistical model assumptions necessary for least squares not satisfied
  • Residuals not IID (Independent Identically Distributed)
harmonic oscillator model residual analysis1
Harmonic Oscillator Model:Residual Analysis
  • Assumptions for Least Squares:
    • Mean of error = 0
    • Variance of error = σ2
    • Covariance of error = 0
    • Residuals IID (Independent Identically Distributed)
  • Segments are not consistent
  • Variance of residuals not constant over time
  • A time pattern is involved, so the covariance is not really zero
  • Amplitude compounds future error—results depend on past error
  • Regular pattern in residual plots
    • Should be random noise, but the residuals are too organized
beam model model fit
Beam Model:Model Fit
  • The beam model is a more accurate fit to the data
beam model zoomed fit
Beam Model:Zoomed Fit
  • Even at the beginning (the area of greatest deviation for the harmonic oscillator model), the beam model follows the data closely
beam model analysis
Beam Model: Analysis
  • Residual comparison
  • Bimodal vs. one mode
  • Better fit
  • Our parameters are a better estimate because Ralph gave us our starting q.
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