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

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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

Harmonic Oscillator Model:Model Fit


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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