Fig. 1

Constrained Fourier approximation fit the gene expression data accurately. A Two examples of true signals (dotted curve), noisy data (’*’), Fourier approximation (solid) and the spline approximation (red dashed) for frequencies of \(4\pi\) (left) and \(\pi\) (right). Spline approximations follow the noise. B The root mean squared error (RMSE) is significantly (two-samples t-test, \(p<10^{-6}\), \(n=100\)) lower for the Fourier approximation than the spline. Furthermore, C 85% of the trials were accurately approximated (lowest RMSE) by Fourier with first and second harmonics. D Frequency analysis of the Fourier approximations: The error is low for frequencies \(<3\pi\), but increases with frequency. The spline approximation (red) is higher, with its mean (mean RMSE of all frequencies) significantly (\(p<10^{-5}\)) higher than the Fourier. A sustained stimulus, an impulse and a wave-like response with frequencies \(\pi /2\), \(2\pi\) and \(4\pi\), respectively, are depicted above. E Deterioration of the noise reduction methods (expressed by the normalized sum of SSE) as the noise variance \(\sigma ^2\) of the gene expression measurements increases. Fourier algorithm performs better than its counterpart for all variances tested