Measuring Variability in Rest-Activity Rhythms from Actigraphy with Application to Characterizing Symptoms of Depression

  • Robert T. KraftyEmail author
  • Haoyi Fu
  • Jessica L. Graves
  • Scott A. Bruce
  • Martica H. Hall
  • Stephen F. Smagula


The twenty-four hour sleep-wake pattern known as the rest-activity rhythm (RAR) is associated with many aspects of health and well-being. Researchers have utilized a number of interpretable, person-specific RAR measures that can be estimated from actigraphy. Actigraphs are wearable devices that dynamically record acceleration and provide indirect measures of physical activity over time. One class of useful RAR measures are those that quantify variability around a mean circadian pattern. However, current parametric and non-parametric RAR measures used by applied researchers can only quantify variability from a limited or undefined number of rhythmic sources. The primary goal of this article is to consider a new measure of RAR variability: the log-power spectrum of stochastic error around a circadian mean. This functional measure quantifies the relative contributions of variability about a circadian mean from all possibly frequencies, including weekly, daily, and high-frequency sources of variation. It can be estimated through a two-stage procedure that smooths the log-periodogram of residuals after estimating a circadian mean. The development of this measure was motivated by a study of depression in older adults and revealed that slow, rhythmic variations in activity from a circadian pattern are correlated with depression symptoms.


Actigraphy Depression Rest-activity rhythm Spectral analysis Time series Wearable technology 


  1. 1.
    Ancoli-Israel S, Cole R, Alessi C, Chambers M, Moorcroft W, Pollak CP (2003) The role of actigraphy in the study of sleep and circadian rhythms. Sleep 26(3):342–392CrossRefGoogle Scholar
  2. 2.
    Troiano RP, Berrigan D, Dodd KW, Masse LC, Tilert T, McDowell M (2008) Physical activity in the United States measured by accelerometer. Med Sci Sports Exerc 40(1):181–188CrossRefGoogle Scholar
  3. 3.
    Smagula SF (2016) Opportunities for clinical applications of rest-activity rhythms in detecting and preventing mood disorders. Curr Opin Psychiatry 29(6):389CrossRefGoogle Scholar
  4. 4.
    Marler MR, Gehrman P, Martin JL, Ancoli-Israel S (2006) The sigmoidally transformed cosine curve: a mathematical model for circadian rhythms with symmetric non-sinusoidal shapes. Stat Med 25(22):3893–3904MathSciNetCrossRefGoogle Scholar
  5. 5.
    Goldsmith J, Zipunnikov V, Schrack J (2015) Generalized multilevel function-on-scalar regression and principal component analysis. Biometrics 71(2):344–353MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Morris JS, Arroyo C, Coull BA, Ryan LM, Herrick R, Gortmaker SL (2006) Using wavelet-based functional mixed models to characterize population heterogeneity in accelerometer profiles: a case study. J Am Stat Assoc 101(476):1352–1364MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Wang J, Xian H, Licis A, Deych E, Ding J, McLeland J, Toedebusch C, Li T, Duntley S, Shannon W (2011) Measuring the impact of apnea and obesity on circadian activity patterns using functional linear modeling of actigraphy data. J Circadian Rhythm 9(1):11CrossRefGoogle Scholar
  8. 8.
    Xiao L, Huang L, Schrack JA, Ferrucci L, Zipunnikov V, Crainiceanu CM (2015) Quantifying the lifetime circadian rhythm of physical activity: a covariate-dependent functional approach. Biostatistics 16(2):352–367MathSciNetCrossRefGoogle Scholar
  9. 9.
    World Health Organization (2017) Depression fact sheet.
  10. 10.
    Hamilton M (1960) A rating scale for depression. J Neurol Neurosurg Psychiatry 23(1):56–62CrossRefGoogle Scholar
  11. 11.
    Zimmerman M, Martinez JH, Young D, Chelminski I, Dalrymple K (2013) Severity classification on the Hamilton depression rating scale. J Affect Disord 150:384–388CrossRefGoogle Scholar
  12. 12.
    van Someren EJW, Hagebeuk EEO, Lijzenga C, Scheltens P, de Rooij SEJA, Jonker C, Pot AM, Mirmiran M, Swaab DF (1996) Circadian rest-activity rhythm disturbances in Alheimer’s disease. Biol Psychiatry 40(4):259–270CrossRefGoogle Scholar
  13. 13.
    Bates DM, Watts DG (2007) Nonlinear regression analysis and its applications, 2nd edn. Wiley, New YorkzbMATHGoogle Scholar
  14. 14.
    Gonçalves BSB, Cavalcanti PRA, Tavares GR, Campos TF, Araujo JF (2014) Nonparametric methods in actigraphy: an update. Sleep Sci 7(3):158–164CrossRefGoogle Scholar
  15. 15.
    Shumway R, Stoffer D (2011) Time series analysis and its applications. Springer, New YorkCrossRefzbMATHGoogle Scholar
  16. 16.
    Wahba G (1980) Automatic smoothing of the log-periodogram. J Am Stat Assoc 75(369):122–132CrossRefzbMATHGoogle Scholar
  17. 17.
    Moulin P (1994) Wavelet thresholding techniques for power spectrum estimation. IEEE Trans Signal Process 42(11):3126–3136CrossRefGoogle Scholar
  18. 18.
    Whittle P (1953) Estimation and information in stationary time series. Ark Mat 2(23):423–434MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Pawitan Y, O’Sullivan F (1994) Nonparametric spectral density estimation using penalized Whittle likelihood. J Am Stat Assoc 89:600–610MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Qin L, Wang Y (2008) Nonparametric spectral analysis with applications to seizure characterization using EEG time series. Ann Appl Stat 2:1432–1451MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Gu C (2013) Smoothing spline ANOVA models, 2nd edn. Springer, New YorkCrossRefzbMATHGoogle Scholar
  22. 22.
    Krafty RT, Collinge WO (2013) Penalized multivariate Whittle likelihood for power spectrum estimation. Biometrika 100(2):447–458MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Cacioppo JT, Tassinary LG, Bernston G (2007) Handbook of psycholphysiology, 3rd edn. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  24. 24.
    Besse P, Ramsay JO (1986) Principal components analysis of sampled functions. Psychometrika 51:285–311MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Rice JA, Silverman BW (1991) Estimating the mean and covariance structure nonparametrically when the data are curves. J R Stat Soc B 53:233–243MathSciNetzbMATHGoogle Scholar
  26. 26.
    Chen K, Lei J (2015) Localized functional principal components analysis. J Am Stat Assoc 110:1266–1275MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Wood SN (2011) Fast stable restricted maximum likelihood and marginal likelihood estimation of semiparametric generalized linear models. J R Stat Soc B 73(1):3–36MathSciNetCrossRefGoogle Scholar
  28. 28.
    Parzen E (1962) Spectral analysis of asymptotically stationary time series. Bulletin de International de Statistique, 33rd Session, PariszbMATHGoogle Scholar
  29. 29.
    Lomb NR (1976) Least-squares frequency analysis of unequally spaced data. Astrophys Space Sci 39:447–462CrossRefGoogle Scholar
  30. 30.
    Krafty RT, Zhao M, Buysse DJ, Thayer JF, Hall MH (2014) Nonparametric spectral analysis of heart rate variability through penalized sum of squares. Stat Med 33(8):1383–1394MathSciNetCrossRefGoogle Scholar
  31. 31.
    Israel B, Buysse DJ, Krafty RT, Begley A, Miewald M, Hall MH (2012) Short-term stability of sleep and heart rate variability in good sleepers and patients with insomnia: for some measures, one night is enough. Sleep 35(9):1285–1291CrossRefGoogle Scholar
  32. 32.
    Dahlhaus R (2012) Locally stationary processes. Handb Stat 30:351–413CrossRefGoogle Scholar

Copyright information

© International Chinese Statistical Association 2019

Authors and Affiliations

  1. 1.Department of BiostatisticsUniversity of PittsburghPittsburghUSA
  2. 2.Department of EpidemiologyUniversity of PittsburghPittsburghUSA
  3. 3.Department of StatisticsGeorge Mason UniversityFairfaxUSA
  4. 4.Department of PsychiatryUniversity of PittsburghPittsburghUSA

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