1. Team 5 Binge Thinkers (formerly known as People doing Math) Statistical Analysis of Vibrating Beam Peter Gross Keri Rehm Regal Ferrulli Anson Chan

2. Overview of Problem Data from a Vibrating Beam Experiment needs to be fitted to a model Dependent Variable: Displacement Independent Variable: Time Initial displacement is caused by voltage induced through patch on beam. 10 data sets were collected One measurement of displacement per microsecond

3. Underview of Solution OK, ….., let’s go on

4. First Modeling Choice The Simple Harmonic Oscillator Solution: Task: Estimate Parameters C and K Nonlinear Regression needed, since partial derivative of regression equation depends on parameters

5. Primary Model Evaluation Model generally portrays the dynamic of the beam’s displacement while vibrating Drawbacks: – Model dampens faster than actual beam does – Model does not even allow for: increasing amplitude over time regardless of which parameters are used random increases and decreases in amplitude

6. Model Extension Since quadratic cost function gave too much weight to spikes in the beginning, the fit was not satisfactory. Next step: Use an absolute cost function instead, which yielded a slightly better fit.

9. Residuals Residual plot for least squares slightly worse than for absolute deviation Residuals are dependent over time (Correlation over time) Possible Heteroskedasticity (needs more testing) Possibly Non-Normal distribution of error term (requires quantile-quantile plot to verify)

12. Summary Statistics of Parameter Estimates

13. Then, uhmmmmm..... Yeaaaaaah Continuing On

14. Further Extension: Cubic Splines We used this for fitting our data points to a smooth curve to compare our previous model with. Allows for more complexity in the structure of our data. Reduces the residuals, which decreases the error term and gives the model more explanatory power.

15. Procedure Modified pre-existing code to handle our data and ran fminsearch to estimate parameters that minimized least squares for all 10 data sets. Used the average of these to obtain an estimate of the true population parameters

17. Conclusions Best fit we were able to found was based on the following parameters: Q = [.2157, 6e-5, 1.3e-5, 2e-4, 0, 0, 0] Model still imperfect, but good approximation to physical system Handles noise significantly better than simple exponentially decaying sinusoidal model Thanks a lot SAMSI Team for an edutational week!