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Autoregressive latent trajectory (ALT) models combine features of latent growth curve models and autoregressive models into a single modeling framework. The development of ALT models has focused primarily on models with linear growth components, but some social processes follow nonlinear trajectories. Although it is straightforward to extend ALT models to allow for some forms of nonlinear trajectories, the identification status of such models, approaches to comparing them with alternative models, and the interpretation of parameters have not been systematically assessed. In this paper we focus on two forms of nonlinear autoregressive latent trajectory (NLALT) models. The first form allows for a quadratic growth trajectory, a popular form of nonlinear latent growth curve models. The second form derives from latent basis models, or freed loading models, that allow for arbitrary growth processes. We discuss details concerning parameterization, model identification, estimation, and testing for the two forms of NLALT models. We include a simulation study that illustrates potential biases that may arise from fitting alternative models to data derived from an autoregressive process and individual-specific nonlinear trajectories. In addition, we include an extended empirical example modeling growth trajectories of weight from birth through age 2.

Two desirable properties of maximum likelihood-based parameter estimates are that the estimates are asymptotically unbiased and asymptotically normally distributed. In this article, the authors test whether the asymptotic properties of maximum likelihood estimation are achieved in sample sizes typically used in applications of group-based trajectory modeling. Through empirical results generated by resampling of population data, they find that the maximum likelihood estimates obtained in group-based trajectory models still provide reasonably close estimates of their true population values and have approximately normal distributions, even when estimated with a sample size as small as n = 500. Furthermore, and more important for the users of these types of models, the authors find similarly good performance in the model's ability to estimate the transformed quantities of main interest: the group trajectories and mixing probabilities.

Group-based trajectory models are used to investigate population differences in the developmental courses of behaviors or outcomes. This note introduces a new Stata command, traj, for fitting to longitudinal data finite (discrete) mixture models designed to identify clusters of individuals following similar progressions of some behavior or outcome over age or time. Normal, Censored normal, Poisson, Zero-inflated Poisson, and Logistic distributions are supported.

The group-based latent trajectory model allows investigators to identify distinctive trajectory groups based on relevant outcomes. Dual trajectory analysis enables examination of the interrelationship between two outcomes simultaneously. This paper describes the application of the group-based dual trajectory model in mental health and substance abuse research. We analyzed two related recovery outcomes: 1) social contact with non-substance abusing friends and 2) stage of substance abuse treatment. We identified four groups for social contact and four groups for substance abuse treatment. We then examined the interrelationship between the two outcomes across the trajectory groups over 10 years. The two outcomes are positively associated longitudinally and evolve in the same direction, although for specific groups, the two outcomes do not covary. By examining dynamic linkages across all trajectory groups between two longitudinal outcomes, the dual model provides a more comprehensive and realistic understanding of the underlying relationships between the two outcomes under investigation.

Over the past decades, several analytic tools have become available for the analysis of reciprocal relations in a non‐experimental context using structural equation modeling (SEM). The autoregressive latent trajectory (ALT) model is a recently proposed model [BOLLEN and CURRAN Sociological Methods and Research (2004) Vol. 32, pp. 336–383; CURRAN and BOLLEN New Methods for the Analysis of Change (2001) American Psychological Association, Washington, DC], which captures features of both the autoregressive (AR) cross‐lagged model and the latent trajectory (LT) model. The present article discusses strengths and weaknesses and demonstrates how several of the problems can be solved by a continuous‐time version: the continuous‐time autoregressive latent trajectory (CALT) model. Using SEM to estimate the exact discrete model (EDM), the EDM/SEM continuous‐time procedure is applied to a CALT model of reciprocal relations between antisocial behavior and depressive symptoms.

A photochemical trajectory model (PTM), coupled with the Master Chemical Mechanism (MCM) describing the degradation of 139 volatile organic compounds (VOCs) in the troposphere, was developed and used for the first time to simulate the formation of photochemical pollutants at Wangqingsha (WQS), Guangzhou during photochemical pollution episodes between 12 and 17 November, 2007. The simulated diurnal variations and mixing ratios of ozone were in good agreement with observed data (R2=0.80, P<0.05), indicating that the photochemical trajectory model - an integration of boundary layer trajectories, precursor emissions and chemical processing - provides a reasonable description of ozone formation in the Pearl River Delta (PRD) region. Calculated photochemical ozone creation potential (POCP) indices for the region indicated that alkanes and oxygenated organic compounds had relatively low reactivity, while alkenes and aromatics presented high reactivity, as seen in other airsheds in Europe. Analysis of the emission inventory found that the sum of 60 of the 139 VOC species accounted for 92% of the total POCP-weighted emission. The 60 VOC species include C2-C6 alkenes, C6-C8 aromatics, biogenic VOCs, and so on. The results indicated that regional scale ozone formation in the PRD region can be mainly attributed to a relatively small number of VOC species, namely isoprene, ethene, m-xylene, and toluene, etc. A further investigation of the relative contribution of the main emission source categories to ozone formation suggested that mobile sources were the largest contributor to regional O3 formation (40%), followed by biogenic sources (29%), VOC product-related sources (23%), industry (6%), biomass burning (1%), and power plants (1%). The findings obtained in this study would advance our knowledge of air quality in the PRD region, and provide useful information to local government on effective control of photochemical smog in the region. © 2010 Elsevier Ltd.

Curran and Bollen combined two models for longitudinal panel data: the latent growth curve model and the autoregressive model. In their model, the autoregressive relationships are modeled between the observed variables. This is a different model than a latent growth curve model with autoregressive relationships between the disturbances. However, when the autoregressive parameter is invariant over time and lies between-1 and 1, it can be shown that these models are algebraically equivalent. This result can be shown to generalize to the multivariate case. When the autoregressive parameters in the autoregressive latent trajectory model vary over time, the equivalence between the autoregressive latent trajectory model and a latent growth curve model with autoregressive disturbances no longer holds. However, a latent growth curve model with time-varying autoregressive parameters for the disturbances could be considered an interesting alternative to the autoregressive latent trajectory model with time-varying autoregressive parameters.

With the development of science and technology, significant changes have taken place in the mode of modern wars. Application of military operating concepts such as 'surgical precision strike' and 'decapitation strike' make higher demands on precision-strike weapons. Ballistic trajectory correction ammunition is being rapidly developed due to its lucrative combination of low cost, high cost-effectiveness ratio, high damage rate, and applicability of existing inventory ammunition. Ballistic control technology has distinct advantages both in cost saving and improvement of the ammunition operational performance. Nose deflection is a feasible, effective, and fast-response flight control mode. The nose part of a projectile can be deflected at a certain angle relative to the projectile body axis, including a pressure difference between the windward and leeward sides of the warhead and generating the respective aerodynamic control force. In this study, a two-rigid-body trajectory model is established based on the multiple rigid body system theory. The proposed model is used to predict the flight trajectory of a projectile with the deflectable nose. Finally, the nose deflection effect on the ballistic trajectory variation is analyzed. The research results obtained provide the theoretical basis for the development of adaptive control smart ammunition and its engineering applications.