Our Week at Math Camp Abridged

1. Our Week at Math CampAbridged Group 2π = [Erin Groark, Sarah Lynn Joyner, Dario Varela, Sean Wilkoff]

2. Agenda • Harmonic Oscillator Model • Parameter Estimates • Standard Errors • Confidence Intervals • Model Fit • Residual Analysis • Beam Model • Model Fit • Analysis • Comparison

3. 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?

4. Harmonic Oscillator Model:Model Fit

5. Harmonic Oscillator Model:Model Fit Zooms Beginning Middle Area of greatest deviation Area of smallest deviation

6. 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

7. Harmonic Oscillator Model:Residual Analysis • Statistical model assumptions necessary for least squares not satisfied • Residuals not IID (Independent Identically Distributed)

8. 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

9. Beam Model:Model Fit • The beam model is a more accurate fit to the data

10. 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

11. Beam Model: Analysis • Residual comparison • Bimodal vs. one mode • Better fit • Our parameters are a better estimate because Ralph gave us our starting q.