Supplementary Materials for “Estimating Periodicity of
Oscillatory Time Series Through Resampling Techniques”
by Maria J. Costa, Bärbel F. Finkenstädt, Peter D. Gould, Julia Foreman,
Karen Halliday, Anthony Hall, and David A. Rand
Additional plots from simulation studies
The definition of outlier used in this paper is motivated by biological rather than distributional arguments. Hence, the boxplots in this web appendix are constructed in a slightly different way. The three central lines composing the box correspond to the usual 25th, 50th and
75th percentiles of the simulated distribution of log10 (SqE) for both the SR method and the FFTNLLS routine excluding all the so-called outliers. However, the whiskers now correspond to
the observed minimum and maximum of log10 (SqE) associated with period estimates within the
circadian range.
1
NC = 4, n = 30
8
4
4
log (SqE)
0
10
10
log (SqE)
NC = 2, n = 30
8
−4
−8
0
−4
SR
−8
otl−FFT−NLLS
SR
NC = 4, n = 60
8
8
4
4
log (SqE)
0
10
10
log (SqE)
NC = 2, n = 60
−4
−8
0
−4
SR
−8
otl−FFT−NLLS
SR
otl−FFT−NLLS
NC = 4, n = 240
8
4
4
log (SqE)
0
10
10
log (SqE)
NC = 2, n = 240
8
−4
−8
otl−FFT−NLLS
0
−4
SR
−8
otl−FFT−NLLS
SR
otl−FFT−NLLS
Supplementary Figure 1. Boxplots of log10 (SqE) for period estimates obtained from synthetic mRNA dynamics for both the SR and the FFT-NLLS methods. In all the plots the
dotted line corresponds to the 50th percentile of the distribution of log10 (SqE) for the SR
method. The crosses represent values of log10 (SqE) associated with outliers.
2
mRNA level
Mild Asymmetry
Severe Asymmetry
600
600
400
400
400
200
200
200
0
Transcription
Moderate Asymmetry
600
0
50
100
0
0
50
100
0
300
300
300
200
200
200
100
100
100
0
0
50
100
Time (hours)
0
0
50
100
Time (hours)
0
0
0
50
100
50
100
Time (hours)
Supplementary Figure 2. Dynamics of synthetic mRNA (top panels) and corresponding transcription functions (bottom panels) for the different asymmetry levels. From left
to right, mild asymmetry level, moderate asymmetry level, and severe asymmetry level.
Markers represent synthetic hourly mRNA observations.
3
Moderate Asymmetry
4
0
0
10
log (SqE)
4
−4
−4
−8
−8
SR
otl−FFT−NLLS
SR
otl−FFT−NLLS
Severe Asymmetry
4
log10(SqE)
log10(SqE)
Mild Asymmetry
0
−4
−8
SR
otl−FFT−NLLS
Supplementary Figure 3. Boxplots of log10 (SqE) for period estimates obtained from synthetic mRNA dynamics satisfying all three levels of asymmetry using both the SR and
FFT-NLLS methods. In all the plots the dotted line corresponds to the 50th percentile of
the distribution of log10 (SqE) for the SR method. Crosses represent values of log10 (SqE)
associated with outliers.
4
mRNA level
Mild Shoulder
Severe Shoulder
400
400
300
300
300
200
200
200
100
100
100
0
τ(t)
Moderate Shoulder
400
0
50
100
0
0
50
100
0
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0
0
50
100
0
Time (hours)
0
50
Time (hours)
100
0
0
0
50
100
50
100
Time (hours)
Supplementary Figure 4. Dynamics of synthetic mRNA (top panels) and corresponding
transcription rates τ(t) (bottom panels) for the different shoulder models. From left to right,
mild shoulder level, moderate shoulder level, and severe shoulder level. Markers represent
synthetic hourly mRNA observations.
5
Moderate Shoulder
6
6
4
4
10
log (SqE)
8
2
2
0
0
−2
−2
SR
otl−FFT−NLLS
SR
otl−FFT−NLLS
Severe Shoulder
8
6
log10(SqE)
log10(SqE)
Mild Shoulder
8
4
2
0
−2
SR
otl−FFT−NLLS
Supplementary Figure 5. Boxplots of log10 (SqE) for period estimates obtained from synthetic mRNA dynamics satisfying all three levels of shoulder pattern using both the SR and
FFT-NLLS methods. In all the plots the dotted line corresponds to the 50th percentile of
the distribution of log10 (SqE) for the SR method. Crosses represent values of log10 (SqE)
associated with outliers.
6
ni = 8, νi = 0.05, σr = 0.1
ni = 16, νi = 0.05, σr = 0.1
i
i
T0
0.04
T0
0.04
T1
T1
T2
T2
K.05
0.02
Size discrepancy
Size discrepancy
K.05
0
−0.02
−0.04
0
0.02
0
−0.02
−0.04
0.2
0.4
0.6
Nominal size
0.8
1
0
0.2
ni = 8, νi = 0.08, σr = 0.2
0.4
0.6
Nominal size
0.8
ni = 16, νi = 0.08, σr = 0.2
i
i
T0
0.04
T0
0.04
T1
T1
T2
T2
K.05
0.02
Size discrepancy
Size discrepancy
K.05
0
−0.02
−0.04
0
1
0.02
0
−0.02
−0.04
0.2
0.4
0.6
Nominal size
0.8
1
0
0.2
0.4
0.6
Nominal size
0.8
Supplementary Figure 6. P-value discrepancy plots for hypotheses tests T 0 , T 1 and T 2 for
different parameter combinations. In all cases, ri = 0.5, i = 1, 2. Horizontal lines represent
5% critical value of the Kolmogorov-Smirnov test, K.05 .
7
1
δ = 0.5, n1 = n2 = 32
δ = 1.5, n1 = n2 = 32
1
1
0.8
0.8
T0
T0
T1
T1
T2
Power
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0.05
0.1
Size
0.15
0
0.2
0
0.05
0.1
Size
0.15
δ = 1.5, n1 = 16, n2 = 32
1
0.8
T0
T1
T2
Power
Power
T2
0.6
0.4
0.2
0
0
0.05
0.1
Size
0.15
0.2
Supplementary Figure 7. Estimated power curves for hypotheses tests T 0 , T 1 and T 2
corrected for true size for different values of δ, n1 and n2 . Other parameters in the data
generating process are fixed to νi = 0.08, ri = 0.5, σri = 0.1, i = 1, 2.
8
0.2