RIAA performed well in the conducted simulations. As shown in , a periodic signal (solid line) with amplitude M
= 1 and frequency ωs
is sampled using the bio-like sampling strategy, which applies 16 time points in (0,8] and 8 more time points in (8,16]. Gaussian noise with parameters μ
= 0 and σ
= 0.5 is assumed during microarray experiments. The resulting time-course expression levels (dots), at a total of 24 time points and the sampling time information were treated as inputs to the RIAA algorithm. demonstrates the result of periodogram estimation. In this example, the grid size Δω
was chosen to be 0.065 and a total of 11 amplitudes corresponding to different frequencies were obtained and shown in the spectrum. Using Fisher's test, the peak at the third grid (frequency = 0.195) was found to be significantly large (p
-value = 2.4 × 10−3
), and hence a periodic gene was claimed.
Figure 2 (a) A time-course periodic signal with frequency = 0.2 sampled by the bio-like sampling strategy; 16 time points are assigned to the interval (0,8], and 8 time points are assigned to the interval (8,16]. (b) The periodogram derived using RIAA. The maximum (more ...)
ROC curves strongly illustrate the performance of RIAA. In Figures and , subplots (a)-(b), (c)-(d), (e)-(f), and (g)-(h) refer to the simulations with regular, bio-like, binomially random, and exponentially random sampling strategies, respectively. Additionally, in the left-hand side subplots (a), (c), (e), and (g), nonperiodic signals were simply Gaussian noise with parameters μ
= 0 and σ
= 0.5, while in the right-hand side subplots (b), (d), (f), and (h), nonperiodic signals involve not only the Gaussian noise but also a transcriptional burst and a sudden drop (27
). Periodic signals were generated using (25
) with amplitude M
= 1, c
= 2, and n
= 24. The only difference in simulation settings between Figures and is the frequency of periodic signals; they are ωs
, respectively. As shown in these figures, LS and DLS can perform well as RIAA when the time-course data are regularly sampled, or mildly irregularly sampled; however, when data are highly irregularly sampled, RIAA outperforms the others. The superiority of RIAA over DLS is particularly clear when the signal frequency is small.
Figure 3 The ROC curves derived from simulations with 24 sampling time points, signal amplitude M = 1, ωs = 0.4π, and Gaussian noise μ = 0 and σ = 0.5. Description of subplots is provided in Section 4.
Figure 4 The ROC Curves derived from simulations with 24 sampling time points, signal amplitude M = 1, ωs = 0.1π, and Gaussian noise μ = 0 and σ = 0.5. Description of subplots is provided in Section 4.
illustrates the results of the real data analysis when these three algorithms, namely, the RIAA, LS, and DLS, were applied. On the x-axis, the numbers indicate the thresholds η that we preserved and classified as periodicities among all yeast genes; on the y-axis, the numbers refer to the intersection of η preserved genes and the proposed periodic candidates listed in the benchmark sets. Figures – demonstrate the results derived from dataset alpha when the 113-gene benchmark set, 352-gene benchmark set, and 518-gene benchmark set were applied, respectively. Similarly, Figures – demonstrate the results derived from dataset alpha 38. The RIAA does not result in significant differences in the numbers of intersections when compared to those corresponding to LS and DLS in most of these cases. However, RIAA shows slightly better coverage when the dataset alpha 38 and the 113-gene benchmark set was utilized ().
Figure 5 The intersection of preserved genes and the benchmark sets using RIAA, LS, and DLS algorithms. (a), (b), and (c) reveal the analysis results when dataset alpha was applied. (d), (e), and (f) reveal the analysis results when dataset alpha 38 was applied. (more ...)