Figure of merit (FOM)

Short definition of Figure of merit (FOM)
A measure of how closely the measured shear spectrum follows a model spectrum

This is the common definition for Figure of merit (FOM), but other definitions maybe discussed within the wiki.

The figure of merit, $\mathrm{FOM}$, provides a measure of how closely the measured spectrum follows a model of the shear spectrum. The measurement uncertainty of a shear spectrum appears to be distributed log-normally with a variance equal to

${\sigma }_{\ln \mathrm{\Psi }}^2=\frac{5}{4}{(N_f-N_V)}^{-7/9}$

where $N_f$ is the number of fft-segments used to make the spectral estimate, with 50% overlap and a cosine window, and $N_V$ is the number of vibration and other signals that were used to clean the spectrum of shear using the Goodman coherent noise removal algorithm.

The spectral values used to estimate the variance of shear (and, hence, the rate of dissipation) should be compared to a reference spectrum of shear for the estimated rate of dissipation. Because the standard deviation of a spectrum, ${\sigma }_{\ln \mathrm{\Psi }}$ is fairly large, the choice of a reference spectrum is not crucial for most practical values of $N_f$. The choices for a reference spectrum include the Panchev-Kesich spectrum , or the Nasmyth spectrum , or a universal spectrum based on over 14000 dimensional spectra ,

The chosen reference spectrum must be dimensionalised using the estimated rate of dissipation. The difference of the logarithm of the measured spectrum and the logarithm of the reference spectrum, for the wavenumber range used to estimated the rate of dissipation, is expected to be distributed normally with a standard deviation of ${\sigma }_{\ln \mathrm{\Psi }}$. However, the sample standard deviation (of the differences) uses a finite number of samples (spectral values) and will inevitably differ from the expectation.

If one draws $N_s$ samples from a population that has a standard deviation of 1, then for 97.5% of the draws the samples will have a standard deviation smaller than

$T_S=1+\sqrt{\frac{2}{N_s}}$

and they will have a mean absolute deviation (MAD) smaller than

$T_M=0.8+\sqrt{\frac{1,56}{N_s}}$.

Thus, a measure of the quality of a spectrum is

$F_S=\frac{s_{\ln \mathrm{\Psi }}}{{\sigma }_{\ln \mathrm{\Psi }}}\frac{1}{T_S}$

where $s_{\ln \mathrm{\Psi }}$ is the sample standard deviation. Or, if based on the mean absolute deviation, the quality metric is

$\mathrm{FOM}=\frac{{\mathrm{MAD}}_{\ln \mathrm{\Psi }}}{{\sigma }_{\ln \mathrm{\Psi }}}\frac{1}{T_M}$

where ${\mathrm{MAD}}_{\ln \mathrm{\Psi }}$ is the sample mean absolute deviation. Thus, the expectation is that $\mathrm{FOM}$ and $F_S$ will be smaller than 1 for 97.5% of the spectra if the measurement quality is good. Values significantly larger than 1 should be flagged, and not used for dissipation estimation. We recommend using $\mathrm{FOM}$ and denote it as the figure of merit.

The various spectral models differ by about 15% and, therefore, an appropriate recommended cutoff for rejection is $\mathrm{FOM}$ = 1.15, and good practice requires that the threshold be explicitly identified for an estimate of the rate of dissipation.

References

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