Critical Limits

FlexPro exclusively uses peak-based critical limit significance levels. This type of confidence limit is of particular merit in ascertaining the significance of the largest spectral component. In this type of test, one seeks to disprove the null hypothesis where one postulates either a white noise signal.

It is important to understand the difference between the more traditional confidence limit and the peak-type critical limit used within FlexPro. A 95% critical limit is that level where in only 1 of 20 separate random noise signals would the largest peak present achieve this height strictly due to random chance. A 99.9% critical limit is similarly that level where in only 1 of 1000 separate random noise sets would the largest peak attain this height.

The traditional confidence or significance level applies to a single set. For example, a standard 95% confidence limit would specify that level where 5% of the points in a single spectrum would be expected to lie above this height strictly due to random chance. This type of significance limit is not used within FlexPro.

Levels

For frequency spectra, five different levels are plotted: 50%, 90%, 95%, 99%, and 99.9%.

Monte Carlo Approximations

The peak-type critical limits were generated using extensive Monte Carlo trials with the algorithms exactly as implemented within FlexPro. The Monte Carlo data were then fitted to effective parametric models. For those cases where the data size n is the only factor, univariate parametric models are used. When an additional factor influences significance, such as an adjustable data window or wavelet parameter, bivariate models are used. For the segmented FFT, where segment size and overlap are additional influences, trivariate Chebyshev polynomials are implemented.

Determining Significance

When the largest peak in a spectrum attains the level of only a 50% critical limit, there is a 50/50 probability the peak could have arisen strictly from chance. On the other hand, when the largest peak exceeds a 99.9% critical limit, there is less than a 1 in 1000 probability the peak arose from chance.