ABSTRACT
Long-term temporal correlations in time series in a form of an event sequence have been characterized using an autocorrelation function that often shows a power-law decaying behavior. Such scaling behavior has been mainly accounted for by the heavy-tailed distribution of interevent times, i.e., the time interval between two consecutive events. Yet, little is known about how correlations between consecutive interevent times systematically affect the decaying behavior of the autocorrelation function. Empirical distributions of the burst size, which is the number of events in a cluster of events occurring in a short time window, often show heavy tails, implying that arbitrarily many consecutive interevent times may be correlated with each other. In the present study, we propose a model for generating a time series with arbitrary functional forms of interevent time and burst size distributions. Then, we analytically derive the autocorrelation function for the model time series. In particular, by assuming that the interevent time and burst size are power-law distributed, we derive scaling relations between power-law exponents of the autocorrelation function decay, interevent time distribution, and burst size distribution. These analytical results are confirmed by numerical simulations. Our approach helps to rigorously and analytically understand the effects of correlations between arbitrarily many consecutive interevent times on the decaying behavior of the autocorrelation function.
ABSTRACT
BACKGROUND: Promotion of facial cleanliness is recommended for the elimination of blinding trachoma, largely because of observational studies that have found an association between various measures of facial uncleanliness and trachoma. However, when a field grader assesses both facial cleanliness and trachoma, associations may be biased. Assessment of photographs of the face and conjunctiva by masked graders may provide a less biased estimate of the relationship between facial cleanliness and trachoma. METHODS: Face photographs, conjunctival photographs, and conjunctival swabs were obtained on a random sample of 0-9-year-old children from each of 40 communities in Amhara region, Ethiopia. Face photographs were assessed for the presence of seven measures of an unclean face (i.e., wet nasal discharge, dry nasal discharge, wet ocular discharge, dry ocular discharge, food, dust/dirt, and flies) by three independent masked photo-graders. Conjunctival photographs were similarly graded in a masked fashion for signs of clinically active trachoma. Conjunctival swabs were processed for Chlamydia trachomatis DNA. RESULTS: Of 2073 children with complete data, 808 (39%) had evidence of clinically active trachoma, 150 (7%) had evidence of ocular chlamydia infection, and 2524 (91%) had at least one measure of an unclean face. Dry ocular discharge had the strongest association with clinically active trachoma (age- and sex-adjusted prevalence ratio [PR] 1.4, 95% CI 1.2-1.6) and ocular chlamydia infection (PR 1.9, 95%CI 1.3-2.9), although significant associations were observed between each of the measures of facial uncleanliness and trachoma. CONCLUSIONS: Masked assessment of face and conjunctival photographs confirmed prior observational studies that have noted associations between various measures of facial uncleanliness and trachoma. The causal relationship between facial uncleanliness and trachoma is unclear since many features used to measure facial cleanliness (e.g., ocular discharge, nasal discharge, and flies) could be consequences of antecedent ocular chlamydia infection. TRIAL REGISTRATION: NCT02754583, clinicaltrials.gov.