1. Our Week at Math Camp Abridged 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 deviationArea 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 = σ 2 –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.