Our Week at Math CampAbridged

Group 2π =

[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

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 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

Beam Model: Analysis•Residual comparison

•Bimodal vs. one mode

•Better fit

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