Akaike causality in state space

We present a new approach of explaining instantaneous causality in multivariate fMRI time series by a state space model. A given single time series can be divided into two noise-driven processes, a common process shared among multivariate time series and a specific process refining the common process. By assuming that noises are independent, a causality map is drawn using Akaike noise contribution ratio theory. The method is illustrated by an application to fMRI data recorded under visual stimulation.     

Granger causality between multiple interdependent neurobiological time series: blockwise versus pairwise methods

Granger causality is becoming an important tool for determining causal relations between neurobiological time series. For multivariate data, there is often the need to examine causal relations between two blocks of time series, where each block could represent a brain region of interest. Two alternative methods are available. In the pairwise method, bivariate autoregressive models are fit to all pairwise combinations involving one time series from the first block and one from the second. The total Granger causality between the two blocks is then derived by summing pairwise causality values from each of these models. This approach is intuitive but computationally cumbersome. Theoretically, a more concise method can be derived, which we term the blockwise Granger causality method. In this method, a single multivariate model is fit to all the time series, and the causality between the two blocks is then computed from this model. We compare these two methods by applying them to cortical local field potential recordings from monkeys performing a sensorimotor task. The obtained results demonstrate consistency between the two methods and point to the significance potential of utilizing Granger causality analysis in understanding coupled neural systems.     

Modeling seasonality in space‐time infectious disease surveillance data

Infectious disease data from surveillance systems are typically available as multivariate times series of disease counts in specific administrative geographical regions. Such databases are useful resources to infer temporal and spatiotemporal transmission parameters to better understand and predict disease spread. However, seasonal variation in disease notification is a common feature of surveillance data and needs to be taken into account appropriately. In this paper, we extend a time series model for spatiotemporal surveillance counts to incorporate seasonal variation in three distinct components. A simulation study confirms that the different types of seasonality are identifiable and that a predictive approach suggested for model selection performs well. Application to surveillance data on influenza in Southern Germany reveals a better model fit and improved one‐step‐ahead predictions if all three components allow for seasonal variation.     

Bayesian analysis of matrix normal graphical models

We present Bayesian analyses of matrix-variate normal data with conditional independencies induced by graphical model structuring of the characterizing covariance matrix parameters. This framework of matrix normal graphical models includes prior specifications, posterior computation using Markov chain Monte Carlo methods, evaluation of graphical model uncertainty and model structure search. Extensions to matrix-variate time series embed matrix normal graphs in dynamic models. Examples highlight questions of graphical model uncertainty, search and comparison in matrix data contexts. These models may be applied in a number of areas of multivariate analysis, time series and also spatial modelling.     

Estimation of growth regulation in natural populations by extended family of growth curve models with fractional order derivative: Case studies from the global population dynamics database

Estimating the trend in population time series data using growth curve models is a central idea in population ecology. Several models, mainly governed by differential or difference equations, have been applied to real data sets to identify general growth pattern and make predictions. In this article, we analyze ecological time series data by fitting mathematical models governed by fractional differential equations (FDE). The order of the FDE (α) is used to quantify the evidence of memory in the population processes. The application of FDE is exemplified by analyzing time series data on two bird species Phalacrocorax carbo (Great cormorant) and Parus bicolor (Tufted titmouse) and two mammal species Castor canadensis (Beaver) and Ursus americanus (American black bear) extracted from the global population dynamics database. Five different population growth models were fitted to these data; density-independent exponential, negative density-dependent logistic and θ-logistic model, positive density-dependent exponential Allee and strong Allee model. Both ordinary and fractional derivative representations of these models were fitted to the time series data. Markov chain Monte Carlo (MCMC) method was used to estimate the model parameters and Akaike information criterion was used to select the best model. By estimating the return rate for each of the time series, we have shown that populations governed by FDE with a small value of α (high level of memory) return to the stable equilibrium faster. This demonstrates a synergistic interplay between memory and stability in natural populations.     

A better way to estimate population trends

Estimation of a population trend from a time series of abundance data is an important task in ecology, yet such estimation remains logistically and conceptually challenging in practice. First, the extent to which unequal intervals in the time series, due to missing observations or irregular sampling, compromise trend estimation is not well‐known. Furthermore, the predominant trend estimation method (loglinear regression of abundance data against time) ignores the possibility of process noise, while an alternative method (the ‘diffusion approximation’) ignores observation error in the abundance data. State‐space models that account for both process noise and observation error exist but have been little used. We study an adaptation of the exponential growth state‐space (EGSS) model for use with missing data in the time series, and we compare its trend estimation to the status quo methods. The EGSS model provides superior estimates of trend across wide ranges of time series length and sources of variation. The performance of the EGSS model even with half of the counts in the time series missing implies that trend estimates may be improved by diverting effort away from annual monitoring and towards increasing time series length or improving precision of the abundance estimates for years that data are collected.     

鞍山西郊水源地地下水水动态初步研究

运用时间序列分析方法,选取月抽水量及月观测静水位,建立时间序列模型。对鞍山市西郊水源地地下水水位进行了拟合,与实测数据相吻合,并进行了预报．     

WGCNA: an R package for weighted correlation network analysis

Background

Modelling the time-related behaviour of biological systems is essential for understanding their dynamic responses to perturbations. In metabolic profiling studies, the sampling rate and number of sampling points are often restricted due to experimental and biological constraints.

Results

A supervised multivariate modelling approach with the objective to model the time-related variation in the data for short and sparsely sampled time-series is described. A set of piecewise Orthogonal Projections to Latent Structures (OPLS) models are estimated, describing changes between successive time points. The individual OPLS models are linear, but the piecewise combination of several models accommodates modelling and prediction of changes which are non-linear with respect to the time course. We demonstrate the method on both simulated and metabolic profiling data, illustrating how time related changes are successfully modelled and predicted.

Conclusion

The proposed method is effective for modelling and prediction of short and multivariate time series data. A key advantage of the method is model transparency, allowing easy interpretation of time-related variation in the data. The method provides a competitive complement to commonly applied multivariate methods such as OPLS and Principal Component Analysis (PCA) for modelling and analysis of short time-series data.     

Extinction risk of a density-dependent population estimated from a time series of population size

Environmental threats, such as habitat size reduction or environmental pollution, may not cause immediate extinction of a population but shorten the expected time to extinction. We develop a method to estimate the mean time to extinction for a density-dependent population with environmental fluctuation. We first derive a formula for a stochastic differential equation model (canonical model) of a population with logistic growth with environmental and demographic stochasticities. We then study an approximate maximum likelihood (AML) estimate of three parameters (intrinsic growth rate r, carrying capacity K, and environmental stochasticity sigma(2)(e)) from a time series of population size. The AML estimate of r has a significant bias, but by adopting the Monte Carlo method, we can remove the bias very effectively (bias-corrected estimate). We can also determine the confidence interval of the parameter based on the Monte Carlo method. If the length of the time series is moderately long (with 40-50 data points), parameter estimation with the Monte Carlo sampling bias correction has a relatively small variance. However, if the time series is short (less than or equal to 10 data points), the estimate has a large variance and is not reliable. If we know the intrinsic growth rate r, however, the estimate of K and sigma(2)(e)and the mean extinction time T are reliable even if only a short time series is available. We illustrate the method using data for a freshwater fish, Japanese crucian carp (Carassius auratus subsp.) in Lake Biwa, in which the growth rate and environmental noise of crucian carp are estimated using fishery records.     

长苞铁杉种群个体年龄与胸径的多维时间序列模型研究

 利用多维时间序列分析方法,建立长苞铁杉(Tsuga longibracteata)种群胸径生长的多维时间序列模型,即Yt=1.7870785Yt-1-0.8950167Yt-2+0.4509997Ut-0.8036035Ut-1+0.3950577Ut-2,其中Yt、Yt-1、Yt-2分别为t年、t-5年、t-10年长苞铁杉胸径值(cm),Ut、Ut-1、Ut-2分别为t年、t-5年、t-10年长苞铁杉种群个体年龄（a）,模型相关系数为0.9998。以年轮确定种群个体年龄法与多维时间序列模型相结合,建立能     

On construction of stochastic genetic networks based on gene expression sequences

Reconstruction of genetic regulatory networks from time series data of gene expression patterns is an important research topic in bioinformatics. Probabilistic Boolean Networks (PBNs) have been proposed as an effective model for gene regulatory networks. PBNs are able to cope with uncertainty, corporate rule-based dependencies between genes and discover the sensitivity of genes in their interactions with other genes. However, PBNs are unlikely to use directly in practice because of huge amount of computational cost for obtaining predictors and their corresponding probabilities. In this paper, we propose a multivariate Markov model for approximating PBNs and describing the dynamics of a genetic network for gene expression sequences. The main contribution of the new model is to preserve the strength of PBNs and reduce the complexity of the networks. The number of parameters of our proposed model is O(n2) where n is the number of genes involved. We also develop efficient estimation methods for solving the model parameters. Numerical examples on synthetic data sets and practical yeast data sequences are given to demonstrate the effectiveness of the proposed model.     

A Variance Components Model for Analyzing Repeated Measures Designs with Non-Stationary Error Structure

When data are collected in the form of multiple measurements on several subjects, they are often analyzed as repeated measures data with some stationary error structure assumed for the errors. For data with non-stationary error structure, the multivariate model is often used. The multivariate model imposes restrictions that are often not met in practice by data of such type. At the same time, they ignore valuable information in the data that are related to time dependencies and time relations. In this paper, we propose a model that is a reparametrization of the multivariate model and is suitable to analyze general repeated measures designs with non-stationary error structure. The model is shown to be a variance components model whose components are estimated using the method of maximum likelihood. Several other properties of the model are derived and discussed including tests of significance. Finally, an example on neurological data is included to demonstrate its application in biological sciences.     

Merging the Idiographic Filter With Dynamic Factor Analysis to Model Process

To model processes we propose merging idiographic filter measurement with dynamic factor analysis. This involves testing whether or not the same latent dynamics (concurrent and lagged factor interrelations) can describe different individuals' observed multivariate time series. The methodology allows fitting, across different individuals, dynamic factor models that are invariant with respect to the latent dynamics, but not necessarily the factor loadings (measurement model). This methodology allows the same latent process to manifest differently from one individual to another, thus recognizing that the process is general but its realization in a given person is to some degree idiosyncratic. The approach is illustrated with empirical data.     

Using Self-Organising Maps (SOMs) to assess synchronies: an application to historical eucalypt flowering records

Self-Organising Map (SOM) clustering methods applied to the monthly and seasonal averaged flowering intensity records of eight Eucalypt species are shown to successfully quantify, visualise and model synchronisation of multivariate time series. The SOM algorithm converts complex, nonlinear relationships between high-dimensional data into simple networks and a map based on the most likely patterns in the multiplicity of time series that it trains. Monthly- and seasonal-based SOMs identified three synchronous species groups (clusters): E. camaldulensis, E. melliodora, E. polyanthemos; E. goniocalyx, E. microcarpa, E. macrorhyncha; and E. leucoxylon, E. tricarpa. The main factor in synchronisation (clustering) appears to be the season in which flowering commences. SOMs also identified the asynchronous relationship among the eight species. Hence, the likelihood of the production, or not, of hybrids between sympatric species is also identified. The SOM pattern-based correlation values mirror earlier synchrony statistics gleaned from Moran correlations obtained from the raw flowering records. Synchronisation of flowering is shown to be a complex mechanism that incorporates all the flowering characteristics: flowering duration, timing of peak flowering, of start and finishing of flowering, as well as possibly specific climate drivers for flowering. SOMs can accommodate for all this complexity and we advocate their use by phenologists and ecologists as a powerful, accessible and interpretable tool for visualisation and clustering of multivariate time series and for synchrony studies.     

Statistical methods for temporal and space–time analysis of community composition data

This review focuses on the analysis of temporal beta diversity, which is the variation in community composition along time in a study area. Temporal beta diversity is measured by the variance of the multivariate community composition time series and that variance can be partitioned using appropriate statistical methods. Some of these methods are classical, such as simple or canonical ordination, whereas others are recent, including the methods of temporal eigenfunction analysis developed for multiscale exploration (i.e. addressing several scales of variation) of univariate or multivariate response data, reviewed, to our knowledge for the first time in this review. These methods are illustrated with ecological data from 13 years of benthic surveys in Chesapeake Bay, USA. The following methods are applied to the Chesapeake data: distance-based Moran''s eigenvector maps, asymmetric eigenvector maps, scalogram, variation partitioning, multivariate correlogram, multivariate regression tree, and two-way MANOVA to study temporal and space–time variability. Local (temporal) contributions to beta diversity (LCBD indices) are computed and analysed graphically and by regression against environmental variables, and the role of species in determining the LCBD values is analysed by correlation analysis. A tutorial detailing the analyses in the R language is provided in an appendix.     

Hidden Markov Models and Animal Behaviour

This paper proposes the use of hidden Markov time series models for the analysis of the behaviour sequences of one or more animals under observation. These models have advantages over the Markov chain models commonly used for behaviour sequences, as they can allow for time-trend or expansion to several subjects without sacrificing parsimony. Furthermore, they provide an alternative to higher-order Markov chain models if a first-order Markov chain is unsatisfactory as a model. To illustrate the use of such models, we fit multivariate and univariate hidden Markov models allowing for time-trend to data from an experiment investigating the effects of feeding on the locomotory behaviour of locusts (Locusta migratoria).     

Early classification of multivariate temporal observations by extraction of interpretable shapelets

ABSTRACT: BACKGROUND: Early classification of time series is beneficial for biomedical informatics problems suchincluding, but not limited to, disease change detection. Early classification can be oftremendous help by identifying the onset of a disease before it has time to fully take hold. Inaddition, extracting patterns from the original time series helps domain experts to gaininsights into the classification results. This problem has been studied recently using timeseries segments called shapelets. In this paper, we present a method, which we callMultivariate Shapelets Detection (MSD), that allows for early and patient-specificclassification of multivariate time series. The method extracts time series patterns, calledmultivariate shapelets, from all dimensions of the time series that distinctly manifest thetarget class locally. The time series were classified by searching for the earliest closestpatterns. RESULTS: The proposed early classification method for multivariate time series has been evaluated oneight gene expression datasets from viral infection and drug response studies in humans. Inour experiments, the MSD method outperformed the baseline methods, achieving highlyaccurate classification by using as little as 40%-64% of the time series. The obtained resultsprovide evidence that using conventional classification methods on short time series is notas accurate as using the proposed methods specialized for early classification. CONCLUSION: For the early classification task, we proposed a method called Multivariate ShapeletsDetection (MSD), which extracts patterns from all dimensions of the time series. Weshowed that the MSD method can classify the time series early by using as little as40%-64% of the time series' length.     

Reverse Engineering Boolean Networks: From Bernoulli Mixture Models to Rule Based Systems

A Boolean network is a graphical model for representing and analyzing the behavior of gene regulatory networks (GRN). In this context, the accurate and efficient reconstruction of a Boolean network is essential for understanding the gene regulation mechanism and the complex relations that exist therein. In this paper we introduce an elegant and efficient algorithm for the reverse engineering of Boolean networks from a time series of multivariate binary data corresponding to gene expression data. We call our method ReBMM, i.e., reverse engineering based on Bernoulli mixture models. The time complexity of most of the existing reverse engineering techniques is quite high and depends upon the indegree of a node in the network. Due to the high complexity of these methods, they can only be applied to sparsely connected networks of small sizes. ReBMM has a time complexity factor, which is independent of the indegree of a node and is quadratic in the number of nodes in the network, a big improvement over other techniques and yet there is little or no compromise in accuracy. We have tested ReBMM on a number of artificial datasets along with simulated data derived from a plant signaling network. We also used this method to reconstruct a network from real experimental observations of microarray data of the yeast cell cycle. Our method provides a natural framework for generating rules from a probabilistic model. It is simple, intuitive and illustrates excellent empirical results.