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 |
- 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.