Dynamic relations for sparsely sampled Gaussian processes

Abstract

In longitudinal studies, it is common to observe repeated measurements data from a sample of subjects where noisy measurements are made at irregular times, with a random number of measurements per subject. Often a reasonable assumption is that the data are generated by the trajectories of a smooth underlying stochastic process. In some cases one observes multivariate time courses generated by a multivariate stochastic process. To understand the nature of the underlying processes, it is then of interest to relate the values of a process at one time with the value it assumes at another time, and also to relate the values assumed by different components of a multivariate trajectory at the same time or at a specific times selected for each trajectory. In addition, an assessment of these relationships will allow to predict future values of an individual's trajectories.Derivatives of the trajectories are frequently more informative than the time courses themselves, for instance in the case of growth curves. It is then of great interest to study the estimation of derivatives from sparse data. Such estimation procedures permit the study of time-dynamic relationships between derivatives and trajectory levels within the same trajectory and between the components of multivariate trajectories. Reviewing and extending recent work, we demonstrate the estimation of corresponding empirical dynamical systems and demonstrate asymptotic consistency of predictions and dynamic transfer functions. We illustrate the resulting prediction procedures and empirical first order differential equations with a study of the dynamics of longitudinal functional data for the relationship of blood pressure and body mass index.

Induction of CXCR3- and CCR5-associated chemokines during acute hepatitis C virus infection

“…The long term trend is estimated via recent developments in functional data analysis [20–23]. After removing this trend, short-term fluctuations about it are examined through Spearman’s rank correlations.…”

“…Plasma samples collected pre- and post-viral acquisition permitted analysis of dynamic changes in chemokine levels in relationship to changes in HCV RNA and ALT levels. A median of 26 data points were analyzed per patient by the use of recently developed statistical techniques [20–23] that enabled apparent patterns of both long-term and short-term dynamics to be quantified and statistically assessed.…”

Exploratory time varying lagged regression: Modeling association of cognitive and functional trajectories with expected clinic visits in older adults

“…7). The lag t − Δ t is chosen by maximizing the absolute value of the estimator of the two dimensional transfer function β( t, s ) of Mueller and Yang (2010) which is equal to the covariance between the response Y ( t ) and the predictor X ( s ) processes, normalized by the variance of X ( s ): $$\frac{\text{cov}false\{Yfalse(tfalse),Xfalse(sfalse)false\}}{\text{var}false\{Xfalse(sfalse)false\}}$$ with respect to s ≤ t . In applications with homoskedastic predictor processes, this can be interpreted as selecting the lag from the predictor’s past trajectory with the highest absolute correlation with the response.…”

“…Existing useful models such as the lagged VCM (Koru-Sengul et al, 2007; Senturk and Mueller, 2008) requires equidistant time grid for estimation. Also, although the recent approach of Mueller and Yang (2010) based on transfer functions, can handle irregular and infrequent data, it is not practical as the number of predictors increase. Therefore, several significant modeling challenges are addressed in the current work, including data sparsity, non-synchronicity of measurement times and the curse of dimensionality.…”

“…Koru-Sengul et al (2007) also proposed an imputation algorithm to fill occasional missing values in the equidistant grid, although this would be impossible for irregular data where subjects are observed at subject specific observation times. Mueller and Yang (2010) proposed the transfer functions $$Efalse\{Yfalse(tfalse)-{\mu }_{Y}false(tfalse)false|Xfalse(sfalse)false\}=\beta false(t,sfalse)false\{Xfalse(sfalse)-{\mu }_{X}false(sfalse)false\},$$ for jointly Gaussian processes where the transfer function β( t, s ) reflects the effect of a lag ( s < t ) of the predictor process on the value of the response at the current time t . Unlike the prior proposals of lagged varying coefficient models, this general model can be estimated from irregular and infrequent data which may not be observed concurrently.…”