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Table 2 BSA and FFT results from simulated harmonic data with noise and background trends

From: Automated Bayesian model development for frequency detection in biological time series

No. ω e a (%) e p (%) b σ FFT σ BSA σ BSA-NS s-n
1 0.5 1 - - 0.49 0.06 0.5 0 0.5 0.0002 70
2 0.5 10 - - 0.49 0.20 0.5 0.0002 0.5 0.0004 6.5
3 0.5 40 - - 0.49 0.54 0.5 0.0005 0.5 0.0011 1.9
4 0.5 10 10 - 0.49 0.27 0.5 0 0.5 0.0003 4.2
5 0.5 10 40 - 0.49 0.57 0.5 0.0002 0.5 0.0007 2.2
6 0.5 100 40 - 0.49 0.89 0.5 0.0006 0.5 0.0020 0.7
7 0.3, 0.5 10 10 - 0.29, 0.51 0.14 0.3, 0.5 0.0003 0.34 0.0832 1
8 0.5 10 - -t 0 0.15 0.5 0.0002 0.5 0.0002 110
9 0.5 10 - -t2 0 0.19 0.5 0.0002 0.5 0.0002 90
10 0.5 10 - -t3 0.02 0.24 0.5 0.0003 0.5 0.0002 35
  1. Each time series was generated with a sine function of angular frequency, ω, of 0.5 rad/s with a level of noise in amplitude, e a , and phase, e p . In some time series a background trend (b) was included, and in case number 7 an additional sine function of 0.3 rad/s is present. The resulting function was sampled 200 times at an interval of 1 s. Results from FFT are presented in the form of the angular frequency with the highest power, , and the estimated standard deviation, σ FFT . The Bayesian frequency estimate at the maximum posterior point is denoted by and its standard deviation by σ BSA . For comparison, the expectation value of ω and its standard deviation computed using BSA and Nested Sampling (BSA-NS) are denoted by and σ BSA-NS . Values of σ below 10-8 are listed as 0. The estimated signal-to-noise ratio (s-n) from the Bayesian analysis is given in the last column. The BSA and BSA-NS approaches deliver the same results, apart from the case of multiple frequencies in a 1D search of ω (case No. 7), which for BSA-NS leads an intermediate estimate between the frequencies with a higher standard deviation.