Statistical Learning With Time Series Dependence: An Application to Scoring Sleep in Mice

Abstract

We develop methodology that combines statistical learning methods with generalized Markov models, thereby enhancing the former to account for time series dependence. Our methodology can accommodate very general and very long-term time dependence structures in an easily estimable and computationally tractable fashion. We apply our methodology to the scoring of sleep behavior in mice. As methods currently used to score sleep in mice are expensive, invasive, and labor intensive, there is considerable interest in developing high-throughput automated systems which would allow many mice to be scored cheaply and quickly. Previous efforts at automation have been able to differentiate sleep from wakefulness, but they are unable to differentiate the rare and important state of rapid eye movement (REM) sleep from non-REM sleep. Key difficulties in detecting REM are that (i) REM is much rarer than non-REM and wakefulness, (ii) REM looks similar to non-REM in terms of the observed covariates, (iii) the data are noisy, and (iv) the data contain strong time dependence structures crucial for differentiating REM from non-REM. Our new approach (i) shows improved differentiation of REM from non-REM sleep and (ii) accurately estimates aggregate quantities of sleep in our application to video-based sleep scoring of mice. Supplementary materials for this article are available online.

KEY WORDS:

• Categorical
• Classification
• Machine learning
• Markov
• REM
• Sequence

Acknowledgments

This research was supported in part by the National Institutes of Health grants T32 HL07713, P01 AG17628, and MH081491.

Notes

Estimating may or may not be necessary for out-of-sample prediction. When the out-of-sample sequence “continues on” from the in-sample sequence as in Figure 4, the initialization distribution for Y _{T + 1} is simply the row of the transition probability matrix A corresponding to the observed Y_T and thus no estimate of is required. When the out-of-sample sequence is an entirely new sequence, an estimate of is necessary.

We also examined other proper scoring rules such as log loss and exponential loss (Savage 1971; Buja, Stuetzle, and Shen 2005). All yielded results that were qualitatively similar to those presented for squared error loss.

Such transitions are indicative of sleep disorders, incorrectly scored epochs, or DREM. DREM is a direct transition from WAKE to REM that occasionally occurs in some mice. Such episodes occur almost exclusively during the lights on period and are the result of brief awakenings interrupting a sustained period of REM sleep (Fujiki et al. 2009).

Table 2 Summary statistics by conditional sleep state

We used default parameter settings for MALLET. TreeCRF requires discrete data and so our covariates were binned using different numbers of quantiles (5, 10, 25, and 100). TreeCRF also requires that the number of leaves be specified (we tried 8, 16, 32, 64, 128, and 256). The TreeCRF results given are the best case over all these parameter settings (b and l are the corresponding number of bins and leaves).