Wavelet spectral testing: application to nonstationary circadian rhythms. (English) Zbl 1433.62304

Hargreaves, Jessica K.; Knight, Marina I.; Pitchford, Jon W.; Oakenfull, Rachael J.; Chawla, Sangeeta; Munns, Jack; Davis, Seth J.

Ann. Appl. Stat. 13, No. 3, 1817-1846 (2019).

Summary: Rhythmic data are ubiquitous in the life sciences. Biologists need reliable statistical tests to identify whether a particular experimental treatment has caused a significant change in a rhythmic signal. When these signals display nonstationary behaviour, as is common in many biological systems, the established methodologies may be misleading. Therefore, there is a real need for new methodology that enables the formal comparison of nonstationary processes. As circadian behaviour is best understood in the spectral domain, here we develop novel hypothesis testing procedures in the (wavelet) spectral domain, embedding replicate information when available. The data are modelled as realisations of locally stationary wavelet processes, allowing us to define and rigorously estimate their evolutionary wavelet spectra. Motivated by three complementary applications in circadian biology, our new methodology allows the identification of three specific types of spectral difference. We demonstrate the advantages of our methodology over alternative approaches, by means of a comprehensive simulation study and real data applications, using both published and newly generated circadian datasets. In contrast to the current standard methodologies, our method successfully identifies differences within the motivating circadian datasets, and facilitates wider ranging analyses of rhythmic biological data in general.

MSC:

62P10	Applications of statistics to biology and medical sciences; meta analysis
42C40	Nontrigonometric harmonic analysis involving wavelets and other special systems
62M15	Inference from stochastic processes and spectral analysis
62-08	Computational methods for problems pertaining to statistics
62M10	Time series, auto-correlation, regression, etc. in statistics (GARCH)
92B25	Biological rhythms and synchronization

Keywords:

wavelets; spectral decomposition; hypothesis testing; circadian rhythms

Software:

fda (R); WaveThresh4; haarfisz; waveclock; Wavos

References:

[1]	Atkinson, A. D., Hill, R. R., Pignatiello, J. J. Jr., Vining, G. G., White, E. D. and Chicken, E. (2017). Wavelet ANOVA approach to model validation. Simul. Model. Pract. Theory 78 18-27.
[2]	Bell-Pedersen, D., Cassone, V. M., Earnest, D. J., Golden, S. S., Hardin, P. E., Thomas, T. L. and Zoran, M. J. (2005). Circadian rhythms from multiple oscillators: Lessons from diverse organisms. Nat. Rev. Genet. 6 544-556.
[3]	Benjamini, Y. and Hochberg, Y. (1995). Controlling the false discovery rate: A practical and powerful approach to multiple testing. J. Roy. Statist. Soc. Ser. B 57 289-300. · Zbl 0809.62014 · doi:10.1111/j.2517-6161.1995.tb02031.x
[4]	Bujdoso, N. and Davis, S. J. (2013). Mathematical modeling of an oscillating gene circuit to unravel the circadian clock network of Arabidopsis thaliana. Front. Plant Sci. 4 3.
[5]	Cho, H. (2016). A test for second-order stationarity of time series based on unsystematic sub-samples. Stat 5 262-277. [Paging previously given as 1-16]. · Zbl 07848568
[6]	Costa, M. J., Finkenstädt, B., Gould, P. D., Foreman, J., Halliday, K. J., Hall, A. J. W. and Rand, D. A. (2011). Estimating periodicity of oscillatory time series through resampling techniques. Centre for Research in Statistical Methodology, Univ. Warwick.
[7]	Das, S. and Nason, G. P. (2016). Measuring the degree of non-stationarity of a time series. Stat 5 295-305. · Zbl 07848572
[8]	Dusik, V., Senthilan, P. R., Mentzel, B., Hartlieb, H., Wülbeck, C., Yoshii, T., Raabe, T. and Helfrich-Förster, C. (2014). The MAP kinase p38 is part of Drosophila melanogaster’s circadian clock. PLoS Genet. 10 e1004565.
[9]	Edwards, K. D., Akman, O. E., Knox, K., Lumsden, P. J., Thomson, A. W., Brown, P. E., Pokhilko, A., Kozma-Bognar, L., Nagy, F. et al. (2010). Quantitative analysis of regulatory flexibility under changing environmental conditions. Mol. Syst. Biol. 6 424.
[10]	Fan, J. and Lin, S.-K. (1998). Test of significance when data are curves. J. Amer. Statist. Assoc. 93 1007-1021. · Zbl 1064.62525 · doi:10.1080/01621459.1998.10473763
[11]	Foley, J. A., DeFries, R., Asner, G. P., Barford, C., Bonan, G., Carpenter, S. R., Chapin, F. S., Coe, M. T., Daily, G. C. et al. (2005). Global consequences of land use. Science 309 570-574.
[12]	Fryzlewicz, P. (2005). Modelling and forecasting financial log-returns as locally stationary wavelet processes. J. Appl. Stat. 32 503-528.
[13]	Fryzlewicz, P. and Nason, G. P. (2006). Haar-Fisz estimation of evolutionary wavelet spectra. J. R. Stat. Soc. Ser. B. Stat. Methodol. 68 611-634. · Zbl 1110.62121 · doi:10.1111/j.1467-9868.2006.00558.x
[14]	Fryzlewicz, P. and Ombao, H. (2009). Consistent classification of nonstationary time series using stochastic wavelet representations. J. Amer. Statist. Assoc. 104 299-312. · Zbl 1388.62253 · doi:10.1198/jasa.2009.0110
[15]	Fryzlewicz, P., Van Bellegem, S. and von Sachs, R. (2003). Forecasting non-stationary time series by wavelet process modelling. Ann. Inst. Statist. Math. 55 737-764. · Zbl 1047.62085 · doi:10.1007/BF02523391
[16]	Guo, W., Dai, M., Ombao, H. C. and von Sachs, R. (2003). Smoothing spline ANOVA for time-dependent spectral analysis. J. Amer. Statist. Assoc. 98 643-652. · Zbl 1040.62082 · doi:10.1198/016214503000000549
[17]	Harang, R., Bonnet, G. and Petzold, L. R. (2006). WAVOS: A MATLAB toolkit for wavelet analysis and visualization of oscillatory systems. BMC Research Notes 5 163.
[18]	Hargreaves, J. K., Knight, M. I., Pitchford, J. W., Oakenfull, R. J. and Davis, S. J. (2018). Clustering nonstationary circadian rhythms using locally stationary wavelet representations. Multiscale Model. Simul. 16 184-214. · Zbl 1462.62665 · doi:10.1137/16M1108078
[19]	Hargreaves, J. K., Knight, M. I., Pitchford, J. W., Oakenfull, R. J., Chawla, S., Munns, J. and Davis, S. J. (2019). Supplement to “Wavelet spectral testing: application to nonstationary circadian rhythms.” DOI:10.1214/19-AOAS1246SUPP. · Zbl 1433.62304
[20]	Holan, S. H., Wikle, C. K., Sullivan-Beckers, L. E. and Cocroft, R. B. (2010). Modeling complex phenotypes: Generalized linear models using spectrogram predictors of animal communication signals. Biometrics 66 914-924. · Zbl 1202.62153 · doi:10.1111/j.1541-0420.2009.01331.x
[21]	Killick, R., Eckley, I. A. and Jonathan, P. (2013). A wavelet-based approach for detecting changes in second order structure within nonstationary time series. Electron. J. Stat. 7 1167-1183. · Zbl 1337.62269 · doi:10.1214/13-EJS799
[22]	Kon, N., Sugiyama, Y., Yoshitane, H., Kameshita, I. and Fukada, Y. (2015). Cell-based inhibitor screening identifies multiple protein kinases important for circadian clock oscillations. Commun. Integr. Biol. 8 e982405.
[23]	Krzemieniewska, K., Eckley, I. A. and Fearnhead, P. (2014). Classification of non-stationary time series. Stat 3 144-157. · Zbl 07847433
[24]	Leise, T. L., Indic, P., Paul, M. J. and Schwartz, W. J. (2013). Wavelet meets actogram. J. Biol. Rhythms 28 62-68.
[25]	Martinez, J. G., Bohn, K. M., Carroll, R. J. and Morris, J. S. (2013). A study of Mexican free-tailed bat chirp syllables: Bayesian functional mixed models for nonstationary acoustic time series. J. Amer. Statist. Assoc. 108 514-526. · Zbl 06195957 · doi:10.1080/01621459.2013.793118
[26]	McClung, C. R. (2006). Plant circadian rhythms. Plant Cell 18 792-803.
[27]	McKay, J. L., Welch, T. D. J., Vidakovic, B. and Ting, L. H. (2012). Statistically significant contrasts between EMG waveforms revealed using wavelet-based functional ANOVA. J. Neurophysiol. 109 591-602.
[28]	Millar, A. J., Carrington, J. T., Tee, W. V. and Hodge, S. K. (2015). Changing planetary rotation rescues the biological clock mutant lhy cca1 of Arabidopsis thaliana. BioRxiv at Cold Spring Harbor Laboratory.
[29]	Minors, D. S. and Waterhouse, J. M. (2013). Circadian Rhythms and the Human. Butterworth-Heinemann, Oxford.
[30]	Moore, D. S. (2007). The Basic Practice of Statistics 2. WH Freeman, New York.
[31]	Moore, A., Zielinski, T. and Millar, A. J. (2014). Online period estimation and determination of rhythmicity in circadian data, using the BioDare data infrastructure. Methods Mol. Biol. 1158 13-44.
[32]	Morris, J. S. (2015). Functional regression. Ann. Rev. Stat. Appl. 2 321-359.
[33]	Morris, J. S., Baladandayuthapani, V., Herrick, R. C., Sanna, P. and Gutstein, H. (2011). Automated analysis of quantitative image data using isomorphic functional mixed models, with application to proteomics data. Ann. Appl. Stat. 5 894-923. · Zbl 1454.62367 · doi:10.1214/10-AOAS407
[34]	Nason, G. P. (2008). Wavelet Methods in Statistics with R. Use R! Springer, New York. · Zbl 1165.62033
[35]	Nason, G. (2013). A test for second-order stationarity and approximate confidence intervals for localized autocovariances for locally stationary time series. J. R. Stat. Soc. Ser. B. Stat. Methodol. 75 879-904. · Zbl 1411.62259 · doi:10.1111/rssb.12015
[36]	Nason, G. P. and Savchev, D. (2014). White noise testing using wavelets. Stat 3 351-362. · Zbl 07847449
[37]	Nason, G. P. and Stevens, K. (2015). Bayesian wavelet shrinkage of the Haar-Fisz transformed wavelet periodogram. PLoS ONE 10 e0137662.
[38]	Nason, G. P., von Sachs, R. and Kroisandt, G. (2000). Wavelet processes and adaptive estimation of the evolutionary wavelet spectrum. J. R. Stat. Soc. Ser. B. Stat. Methodol. 62 271-292.
[39]	Nicholson, F. A., Smith, S. R., Alloway, B. J., Carlton-Smith, C. and Chambers, B. J. (2003). An inventory of heavy metals inputs to agricultural soils in England and Wales. Sci. Total Environ. 311 205-219.
[40]	Ogden, T. R. (1997). On preconditioning the data for the wavelet transform when the sample size is not a power of two. Comm. Statist. Simulation Comput. 26 467-486. · Zbl 0900.62208 · doi:10.1080/03610919708813391
[41]	Oh, H.-S., Ammann, C. M., Naveau, P., Nychka, D. and Otto-Bliesner, B. L. (2003). Multi-resolution time series analysis applied to solar irradiance and climate reconstructions. J. Atmos. Sol.-Terr. Phys. 65 191-201.
[42]	Perea-García, A., Andrés-Bordería, A., de Andrés, S. M., Sanz, A., Davis, A. M., Davis, S. J., Huijser, P. and Peñarrubia, L. (2015). Modulation of copper deficiency responses by diurnal and circadian rhythms in arabidopsis thaliana. J. Exp. Bot. 67 391-403.
[43]	Price, T. S., Baggs, J. E., Curtis, A. M., FitzGerald, G. A. and Hogenesch, J. B. (2008). WAVECLOCK: Wavelet analysis of circadian oscillation. Bioinformatics 24 2794-2795.
[44]	Priestley, M. B. (1981). Spectral Analysis and Time Series. Vol. 1. Univariate Series, Probability and Mathematical Statistics. Academic Press, New York. · Zbl 0537.62075
[45]	Priestley, M. B. and Subba Rao, T. (1969). A test for non-stationarity of time-series. J. Roy. Statist. Soc. Ser. B 31 140-149. · Zbl 0182.51403 · doi:10.1111/j.2517-6161.1969.tb00775.x
[46]	Ramsay, J. O. and Silverman, B. W. (2005). Functional Data Analysis, 2nd ed. Springer Series in Statistics. Springer, New York. · Zbl 1079.62006
[47]	Senesil, G. S., Baldassarre, G., Senesi, N. and Radina, B. (1998). Trace element inputs into soils by anthropogenic activities and implications for human health. Chemosphere 39 343-377.
[48]	Shumway, R. H. (1988). Applied Statistical Time Series Analysis. Prentice-Hall, Englewood Cliffs, NJ. · Zbl 0663.62061
[49]	Tavakoli, S. and Panaretos, V. M. (2016). Detecting and localizing differences in functional time series dynamics: A case study in molecular biophysics. J. Amer. Statist. Assoc. 111 1020-1035.
[50]	Torrence, C. and Compo, G. P. (1998). A practical guide to wavelet analysis. Bull. Am. Meteorol. Soc. 79 61-78.
[51]	Van Bellegem, S. and von Sachs, R. (2008). Locally adaptive estimation of evolutionary wavelet spectra. Ann. Statist. 36 1879-1924. · Zbl 1142.62067 · doi:10.1214/07-AOS524
[52]	Vidakovic, B. (2001). Wavelet-based functional data analysis: Theory, applications and ramifications. In Proceedings of the 3rd Pacific Symposium on Flow Visualization and Image Processing, Maui, HI.
[53]	Vitaterna, M. H., Takahashi, J. S. and Turek, F. W. (2001). Overview of circadian rhythms. Alcohol Res. Health 25 85-93.
[54]	von Sachs, R. and Neumann, M. H. (2000). A wavelet-based test for stationarity. J. Time Series Anal. 21 597-613. · Zbl 0972.62085 · doi:10.1111/1467-9892.00200
[55]	Zielinski, T., Moore, A. M., Troup, E., Halliday, K. J. and Millar, A. J. (2014). Strengths and limitations of period estimation methods for circadian data. PLoS ONE 9 96462.