An Approximation of a Longitudinal Stochastic Model

Citation: Salah KA (2019) An Approximation of a Longitudinal Stochastic Model. Int J Clin Biostat Biom 5:020. doi.org/10.23937/2469-5831/1510020 Accepted: March 23, 2019: Published: March 25, 2019 Copyright: © 2019 Salah KA. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

time, to understand the effect of the covariates on the response variable, and to understand the within subject correlation structure.Let Where ( ) is an intra subject error.

Introduction
A longitudinal data consists of measurements of a single variable taken repeatedly over time from an individual.Any approach used to analyze such data must properly consider the correlations among the observations, see Li, N., et al. [1].The typical structure of longitudinal data is numerous measurements of a possible multivariate response variable on each subject.There could also be covariates, possibly time varying, that influence the response variable [2].The aim in the analysis of such data is to understand the changes in the mean structure of the response variable with ous white noise with a kernel, whose shape determines the covariance structure of the resulting process.This approach is an alternative to traditional geostatistical techniques, where a covariance function is specified directly, but allows for increased flexibility, since the choice of the kernel also allows for features such as non-stationary, anisotropy, and edge effects.Moreover, model (1.2) is temporal longitudinal model, by applying the proposed approximation, the dimensionality of the complex temporal process significantly reduced.
The model we propose for Where U is a mean zero Gaussian process, and is an approximation of the IOU stochastic process W . Hence, model in (1.2) becomes ( ) ( ) ( ) But rather than specify ( ) Where ( ) The resulting covariance function for ( ) U t depends only on the displacement vector d = t − s and is given by  [7].
In this paper, we introduce an approximation of the IOU process which still gives an efficient inference of model parameters and reducing the dimensionality and complexity of the model.The details of this simple approach are given in the following sections, including an approximate formulation, the likelihood and parameter estimations.The usefulness of this modeling approach is then demonstrated by simulations.

The Stochastic Process and Approximation
The main disadvantage of the IOU process is that it is not stationary; hence it is necessary to have a natural time zero for each individual.In some applications, it may be that there is no natural time zero, or that time zero is not exactly known.Large Longitudinal datasets are often defined on naturally heterogeneous fields or have other inherently spatially varying conditions.Therefore, it is unreasonable to expect a response variable to be well-modeled by a stationary process over a large domain space.However, using non-stationary models is difficult in practice due to the conceptual challenges in specifying the model and the computational challenges of fitting the model when the data is so large that memory constraint prevent formation of the covariance matrix.
We propose to approximate the IOU stochastic process i W into flexible spatial model that can be constructed by convolving a very simple and independent, process with a kernel function.This approximation for constructing a spatial process introduces a number of advantages over specification through a spatial covariogram.In particular, this process convolution specification leads to computational simplifications and easily extends beyond simple stationary models.Our modeling approach is similar to that in Higdon [8], provide simple representations of such model by convolving continu-Table 1: Various kernels and their induced covariance functions in the two-dimensional plane.

Kernel
Covariance function To fit the full model and make inference about the population parameters, Adaptive Rejection Metropolis Sampling (ARMS), and Gibbs and ARMS sampling techniques are used.These methods are a MCMC technique for drawing dependent samples from complex high dimensional distributions, see Waezizadeh, and Mehrpooya [15].The posterior distribution converges was checked by Gelman-Rubin convergence statistic R (posterior consistency), as modified by Brooks and Gelman [16].In order to apply one of these methods on our model, posterior for each parameter must be derived, and then a proposed prior density for each of these parameters must be chosen.Based on the likelihood functions in (3.1) and (3.2), with the notations (IG) denotes for Inverse Gamma, and (N) denotes for Normal, the conditional densities of the unknown model parameters are given as follows: For the error parameter 2 ε σ ( ) ( ) In the same manner we found: The intercept variance 2 a σ : ( ) The random intercept i a , for , where The average rate of the slope b : , where exponential, and spherical covariograms for the process ( ) U t , [9].In addition, the covariogram induced by the biwieght kernel Cleveland [10] is also shown.
The process convolution approach gives an approach to build dependent spatial processes, see Ver Hoef and Barry [11].The basic idea is to build processes ( ) U t that share part of a common latent process in their construction.Perhaps the biggest attraction to these process convolution models is that they give a framework for developing new classes of space and space-time models that allow for more realistic space-time dependence while maintaining some analytic tractability.Generally, one can construct a space-time process by first defining a simple, possibly discrete, process over space and time, and then smoothing it out with one or more kernels, giving a smooth process over space and time.This constructive approach is appealing since the resulting models can be extended to allow for generalizations such as non-stationarity, non-Gaussian models, and non-separable space-time dependence structures.See Wolpert and Ickstadt [12], and Higdon, et al. [13] for some purely spatial applications, and Higdon [14] for a space-time model.In addition, models can be constructed in such a way to facilitate computation -such as restricting the underlying process to reside on a lattice so that fast Fourier transforms can be employed.

The Likelihood and Priors
Based upon our previous assumptions, the unknown parameters in model ( 2 , , , , , Where, [ ] .and . .    denote marginal and conditional densities, respectively, Ω denotes all model unknown parameters.
For the prior density of Ω we assume that From the independency assumptions, the posterior density of all unknown model parameters , and ∑ is given Since all posterior densities are in standard form, then it is easy to choose conjugate priors for all model (2.1) parameters, drawing random variates using Gibbs sampler from their full conditional distributions is straightforward since their full conditional densities are standard distributions.Therefore, we use the full conditional density as proposal density.At each updating step for these parameters, a new draw from the full conditional density is always accepted.

Simulation
To illustrate our proposed model, we setup our simulation study represents a randomize clinical trial, in which M = 500 subjects are randomized.Each longitudinal marker in model (2.1), ( ) was simulated as the sum of the trajectory function X t and the error terms ( ) , each subject has its observed longitudinal measured The effect of the regression parameter on the marker ( )[ ] 0 , 1,..., : ., , where -1  We obtained 3000 iterations after a burn-in of 1000 iterations.Convergence was checked by monitoring histories of sampled quantities with several different starting points.The histogram, the time series plots of one sequence of Gibbs samples for different number of iterations and the average number of these iterations for the parameter a µ are presented in Figure 1.
We also used the Gelman-Rubin convergence statistic, as modified by Brooks and Gelman [15].They emphasize that one should be concerned both with convergence of R is the ratio pooled-width within-width (the ratio of width of the central 80% interval based on pooled runs and the average width of 80% intervals within the individual runs), to one, and with convergence of both the pooled and within interval widths to stability.For our analyses, all R's converged to 1 within 3000 iterations and hence the burn-in of 1000.The analysis for the 500 simulated data sets for a single scenario took approximately 2:30 hours to run the model under the approximate model.While it took almost 6 hours to run the analysis without approximations.Since the calculations for the simulation study were highly computationally intensive, we have used the cluster with about 20 nodes with AMD Quad-Core Opteron 835X, 4 × 2G Hz, and 16 GB RAM per node.
i to the conditional likelihood function is given by

Figure 1 :
Figure 1: Histogram, time series and average values plots respectively for the parameter values µ α at 500, 1000, and 2000 iterations respectively, using Gibbs sampler.
( ) ij t ≥ .Values of ( ) i ij Y t , the observed longitudinal data are taken to follow ( ) ( ) ( )

Table 2 :
Monte Carlo Summary statistics of the parameter estimates.

Table 3 :
Posterior estimates from proposed and IOU models.model, the proposed model results in improved estimates almost for all parameters.The proposed model demonstrated significant reductions in execution time.This approach effectively eliminates the deficiency of non-spatial huge data access by replacing such patterns in hotspots of applications with spatial sites data at a space time t at runtime.The execution time reduced by 60% illustrate the efficiency of proposed model.