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Our Week at Math Camp AbridgedPowerPoint Presentation

Our Week at Math Camp Abridged

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Our Week at Math Camp Abridged

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Our Week at Math CampAbridged

Group 2π =

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

- Harmonic Oscillator Model
- Parameter Estimates
- Standard Errors
- Confidence Intervals

- Model Fit
- Residual Analysis

- Parameter Estimates
- Beam Model
- Model Fit
- Analysis

- Comparison

- 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?

Beginning Middle

Area of greatest deviation

Area of smallest deviation

- 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

- Statistical model assumptions necessary for least squares not satisfied
- Residuals not IID (Independent Identically Distributed)

- 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

- The beam model is a more accurate fit to the data

- Even at the beginning (the area of greatest deviation for the harmonic oscillator model), the beam model follows the data closely

- Residual comparison
- Bimodal vs. one mode
- Better fit
- Our parameters are a better estimate because Ralph gave us our starting q.