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# Our Week at Math Camp Abridged - PowerPoint PPT Presentation

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 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

• Beam Model

• Model Fit

• Analysis

• Comparison

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

Beginning Middle

Area of greatest deviation

Area of smallest deviation

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

• Statistical model assumptions necessary for least squares not satisfied

• Residuals not IID (Independent Identically Distributed)

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

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

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

• Residual comparison

• Bimodal vs. one mode

• Better fit

• Our parameters are a better estimate because Ralph gave us our starting q.