# Forcing Function Estimates as a Function of Modal Damping

In a previous article, a method was described for the estimation of the residuals measured in a resonant system, given a small amount of memory of previous values in the timeseries. The previously supplied formulas are thought to be applicable in single as well as multiple timeseries. The accuracy of estimation is thought to be affected by several parameters in the analysis, and the dynamic behavior of the measurements themselves. In this experiment, we attempt to correlate estimation accuracy with modal damping. My hypothesis was that, on average, the estimation accuracy would be negatively affected by increased damping. This reasoning came from previous work wherein the accuracy of eigenvalue estimation was observed to be much better when those eigenvalues had larger radii, and therefore less damping.

A number of metrics have been compiled in an attempt to correlate the previously described regression model's forcing function estimation accuracy to other factors affecting the analysis. The forcing function estimation accuracy is expressed as the mean magnitude of correlation between the estimated vector and the actual vector. The sample for this experiment consisted of 10,000 randomly generated models, which simulate a noisy signal with modal activity. Because these models are synthesized, their forcing function vectors are known, thus allowing the comparison. The sample is divided into 10 groups of 1,000 samples each, whose eigenvalues range in minimum magnitude from 0.5 to 0.95 . The maximum magnitude is always unity, and the radii are evenly distributed from $0$ to $2\pi$ radians. Trials began with the lowest damping, and gradually increased.

The octave code for this test may be found here. In addition to collecting the mean absolute value correlation between the estimated and actual forcng function, the code also collects detailed information about each of the 10,000 models, for future analysis. In particular, the locations of each model's estimated and actual eigenvalues are stored, as well as a short plot of the first 50 points in the estimated vs actual forcing function. The dataset for the entire experiment may be found here (zip). Be forewarned: this directory is 4.5 G when unzipped!

A graph plotting the trend between trial number (increased modal damping) and estimation accuracy refutes my initial hypothesis: The average estimation accuracy appears to be positively correlated with damping. A few examples might illustrate the finer points of this experiment.

Trial 1:

mics: 1
order: 4
winsize: 10000

forcing function estimation accuracy: 0.7467240384172157

Data detailing a few exemplary cases within this trial are shown below:

Model 1.73: actual eigenvalues (red):
(0.3694800737445258,0.9285568859595529)
(0.3694800737445258,-0.9285568859595529)
(-0.6684398860942371,0.6960931376307926)
(-0.6684398860942371,-0.6960931376307926)

estimated eigenvalues (blue):
(0.3714917366988897,0.9254112183034985)
(0.3714917366988897,-0.9254112183034985)
(-0.6593205717659008,0.6931863769565488)
(-0.6593205717659008,-0.6931863769565488) correlation between actual and estimated forcing function: -0.8400878317364711

Model 1.52 actual eigenvalues (red):
(-0.6374740797145261,0.7662827800105209)
(-0.6374740797145261,-0.7662827800105209)
(-0.6415220398236982,0.7046879734977175)
(-0.6415220398236982,-0.7046879734977175)

estimated eigenvalues (blue):
(-0.6349112252691026,0.7657413061566818)
(-0.6349112252691026,-0.7657413061566818)
(0.2930917014987452,0.4257456302175208)
(0.2930917014987452,-0.4257456302175208) correlation between actual and estimated forcing function: 0.9831361405217255

It is surprising to note the discrepancy between this model's accuracy of eigenvalue estimation, and accuracy of forcing function estimation. In model 1.73, the analysis gave much closer results to the measured eigenvalues, but was significantly less accurate with regards to the residual signal. This sheds some doubt on reasoning behind the hypothesis, which claimed the residual estimation accuracy would negatively correlate with damping, because this was the trend with eigenvalue estimation accuracy.

Trial 2:

mics: 1
order: 4
winsize: 10000

forcing function estimation accuracy: 0.7915352200283716

Data detailing a few exemplary cases within this trial are shown below:

Model 2.11 actual eigenvalues (red):
(-0.2860662474085908,0.9096354558183487)
(-0.2860662474085908,-0.9096354558183487)
(0.5040615050618753,0.8393218506239286)
(0.5040615050618753,-0.8393218506239286)

estimated eigenvalues (blue):
(-0.3059271540916486,0.9243221244613578)
(-0.3059271540916486,-0.9243221244613578)
(0.4632989551481991,0.8442485784854779)
(0.4632989551481991,-0.8442485784854779) correlation between actual and estimated forcing function: -0.4122935368994767

Model 2.12 actual eigenvalues (red):
(-0.9063836964250392,0.1777163451414865)
(-0.9063836964250392,-0.1777163451414865)
(-0.2297625866117844,-0.9442173752500621)
(-0.2297625866117844,0.9442173752500621)

estimated eigenvalues (blue):
(-0.7016093644449533,0.1856952663260466)
(-0.7016093644449533,-0.1856952663260466)
(0.2188297515763993,0.2741817289605008)
(0.2188297515763993,-0.2741817289605008) correlation between actual and estimated forcing function: 0.9764765656574066

Trial 3:

mics: 1
order: 4
winsize: 10000

forcing function estimation accuracy: 0.8157407236336748

Model 3.12 actual eigenvalues (red):
(-0.005721989806571928,0.8830286807456503)
(-0.005721989806571928,-0.8830286807456503)
(0.01924164032056006,0.8662931489264915)
(0.01924164032056006,-0.8662931489264915)

estimated eigenvalues (blue):
(-0.0347117422460286,0.9227119959619106)
(-0.0347117422460286,-0.9227119959619106)
(-0.005126282858746486,0.1613246402015194)
(-0.005126282858746486,-0.1613246402015194) correlation between actual and estimated forcing function: 0.9971928638537638

Model 3.29 actual eigenvalues (red):
(-0.8938052987468124,0.1443901742130357)
(0.9529163060498177,0.2397177481507036)
(-0.8938052987468124,-0.1443901742130357)
(0.9529163060498177,-0.2397177481507036)

estimated eigenvalues (blue):
(-0.0347117422460286,0.9227119959619106)
(-0.0347117422460286,-0.9227119959619106)
(-0.005126282858746486,0.1613246402015194)
(-0.005126282858746486,-0.1613246402015194) correlation between actual and estimated forcing function: 0.9099120725282704

Trial 4:

mics: 1
order: 4
winsize: 10000

forcing function estimation accuracy: 0.8327778979368315

Model 4.5 actual eigenvalues (red):
(-0.3641614230546495,0.5814003778261088)
(-0.3641614230546495,-0.5814003778261088)
(-0.2393574466259,0.6511879883690893)
(-0.2393574466259,-0.6511879883690893)

estimated eigenvalues (blue):
(-0.2769426891829918,0.6363811660486625)
(-0.2769426891829918,-0.6363811660486625)
(0.1382678063414834,0.1799797628203285)
(0.1382678063414834,-0.1799797628203285) correlation between actual and estimated forcing function: 0.9997702210041248

Trial 10:

mics: 1
order: 4
winsize: 10000

forcing function estimation accuracy: 0.880880768075054

Model 10.33 actual eigenvalues (red):
(-0.2498917241045596,0.6744791330014946)
(-0.8728715448737022,0.2365982000016698)
(-0.2498917241045596,-0.6744791330014946)
(-0.8728715448737022,-0.2365982000016698)

estimated eigenvalues (blue):
(-0.7412776635317383,0)
(0.2404571195463273,0)
(-0.06972399635182146,0.5138181681305076)
(-0.06972399635182146,-0.5138181681305076) correlation between actual and estimated forcing function: 0.9686500601384376

Future work will correlate order, number of channels, and eigenvalue estimation accuracy with the forcing function estimation accuracy. Still more will focus on the application of this analysis technique to real-world measurements.