Bayesian modeling and forecasting of vector autoregressive moving average processes

Computed tomography (CT) angiography (CTA) is a non-invasive imaging method used to detect arteries and examine various brain diseases. When CTA is performed for follow-up or postoperative evaluation, reproducibility of vessel delineation is required. A reproducible and stable contrast enhancement can be achieved by manipulating the factors affecting it. Previous studies have investigated several factors that alter the contrast enhancement of arteries. However, no reports establishing the effect of different operators on contrast enhancement exist. To assess the differences between inter-operator arterial contrast enhancement in cerebral CTA using Bayesian statistical modeling. Image data were obtained using a multistage sampling method from the cerebral CTA scans of patients who underwent the process between January 2015 and December 2018. Several Bayesian statistical models were developed, and the objective variable was the mean CT number of the bilateral internal carotid arteries after contrast enhancement. The explanatory variables were sex, age, fractional dose (FD), and the operator's information. The posterior distributions of the parameters were computed via Bayesian inference using the Markov chain Monte Carlo (MCMC) method, with the Hamiltonian Monte Carlo method employed as the algorithm. The posterior predictive distributions were computed using the posterior distributions of the parameters. Finally, the differences between inter-operator arterial contrast enhancement on the CT number in cerebral CTA were estimated. The posterior distributions showed that all parameters representing the difference between operators included zero at the 95% credible intervals (CIs). The maximum mean difference between inter-operator CT number in the posterior predictive distribution was only 12.59 Hounsfield units (HUs). The Bayesian statistical modeling results suggest that contrast enhancement of cerebral CTA examination between operator-to-operator differences in postcontrast CT number was small compared to those within-operator differences resulting from factors not considered in the model.

In this work an algorithm to adjust parameters using a Bayesian method for cumulative rainfall time series forecasting implemented by an ANN-filter is presented. The criterion of adjustment comprises to generate a posterior probability distribution of time series values from forecasted time series, where the structure is changed by considering a Bayesian inference. These are approximated by the ANN based predictor in which a new input is taken in order for changing the structure and parameters of the filter. The proposed technique is based on the prior distribution assumptions. Predictions are obtained by weighting up all possible models and parameter values according to their posterior distribution. Furthermore, if the time series is smooth or rough, the fitting algorithm can be changed to suit, in function of the long or short term stochastic dependence of the time series, an on-line heuristic law to set the training process, modify the NN topology, change the number of patterns and iterations in addition to the Bayesian inference in accordance with Hurst parameter H taking into account that the series forecasted has the same H as the real time series. The performance of the approach is tested over a time series obtained from samples of the Mackey-Glass delay differential equations and cumulative rainfall time series from some geographical points of Cordoba, Argentina.

Estimation of flock/herd-level true Mycobacterium avium subspecies paratuberculosis prevalence on sheep, beef cattle and deer farms in New Zealand using a novel Bayesian model

Approximated sensitivity analysis in posterior predictive distribution

We have discussed various statistical models and their implementation so far. However, the ways we used the MCMC samples from the posterior distributions and predictive distributions were only kept on very basic levels, such as computing the intervals and visualizations. In this chapter, we will introduce more advanced usages of the MCMC sample. They would be helpful in practice because it is very common to encounter the situation where we need to extract more information from the MCMC sample.

In this article we present a Bayesian prediction of multiplicative seasonal autoregressive moving average (SARMA) processes using the Gibbs sampling algorithm. First, we estimate the unobserved errors using the nonlinear least squares (NLS) method to approximate the likelihood function. Second, we employ conjugate priors on the model parameters and initial values and assume the model errors are normally distributed to derive the conditional posterior and predictive distributions. In particular, we show that the conditional posterior distribution of the model parameters and the variance are multivariate normal and inverse gamma respectively, and the conditional predictive distribution of the future observations is a multivariate normal. Finally, we use these closed-form conditional posterior and predictive distributions to apply the Gibbs sampling algorithm to approximate empirically the marginal posterior and predictive distributions, enabling us easily to carry out multiple-step ahead predictions. We evaluate our proposed Bayesian method using simulation study and real-world time series datasets.

Cryptocurrency markets have emerged as a dynamic and intriguing domain, with Bitcoin at the forefront, captivating the attention of investors, researchers, and enthusiasts alike. The volatile nature of Bitcoin prices presents both opportunities and challenges for market participants seeking to understand and anticipate its movements. In this study, we delve into the realm of time series analysis to explore the feasibility of predicting The research journey begins with meticulous data preprocessing steps to ensure the quality and integrity of the input data. Leveraging Python libraries such as pandas and NumPy, we cleanse and format the historical Bitcoin price data, laying the foundation for subsequent analysis. Key preprocessing tasks include handling missing values, normalization, and addressing any anomalies or outliers that may distort the underlying patterns. With the data prepared, our attention turns to assessing the stationarity of the Bitcoin price time series—a fundamental prerequisite for applying classical time series models. Through visual inspection and statistical tests such as the Augmented Dickey-Fuller (ADF) test, we ascertain the presence of trends or seasonality that could influence the modelling process. To mitigate such effects, we employ techniques such as differencing and transformations, including the Box-Cox transformation, to stabilize the variance of the data. Armed with a stationary time series, we embark on the core of our analysis: modelling Bitcoin prices using Autoregressive Integrated Moving Average (ARIMA) and Seasonal Autoregressive Integrated Moving Average with Exogenous Variables (SARIMAX) models. These models, renowned for their versatility and effectiveness in capturing temporal dependencies, offer a sophisticated framework for forecasting time series data. Guided by the principles of parsimony and model selection criteria such as the Akaike Information Criterion (AIC), we systematically explore the parameter space to identify the optimal specifications for our models. The efficacy of the chosen models is rigorously evaluated through diagnostic checks, encompassing residual analysis, model fit statistics, and out-of-sample validation. Insights gleaned from these assessments inform our confidence in the models' predictive capabilities and guide our interpretation of the forecasted outcomes. Finally, armed with a validated model, we turn our gaze to the future, employing it to generate forecasts of Bitcoin prices for forthcoming periods. Visualizations juxtaposing predicted prices against observed values provide a compelling narrative of the model's performance and offer stakeholders valuable insights into potential market trends and dynamics. In summary, this research contributes to the burgeoning field of cryptocurrency analytics by showcasing the application of time-tested statistical methodologies to forecast Bitcoin prices. By leveraging the power of ARIMA and SARIMAX models, we illuminate the intricate patterns underlying Bitcoin's price dynamics, empowering market participants with actionable intelligence for informed decision-making in an ever-evolving landscape. Keywords Cryptocurrency Markets, Bitcoin Price Prediction, Time Series Analysis, Python Programming, Data Preprocessing, Pandas, NumPy, Stationarity Testing, Augmented Dickey-Fuller (ADF) Test, Box-Cox Transformation, ARIMA Model, SARIMAX Model, Model Selection, Akaike Information Criterion (AIC), Diagnostic Checks, Residual Analysis, Out-of-Sample Validation, Forecasting, Visualization, Market Trends, Decision-Making, Cryptocurrency Analytics

The paper takes up posterior analysis of a linear regression model with disturbances from a logarithmic Birnbaum–Saunders distribution. The posterior distribution cannot be analyzed using analytical techniques and, therefore, posterior simulation methods are proposed. Computations are organized around Markov Chain Monte Carlo methods. Associated methods to compute posterior odds and posterior predictive distributions are also discussed.

This paper develops a methodology for approximating the posterior first two moments of the posterior distribution in Bayesian inference. Partially specified probability models that are defined only by specifying means and variances, are constructed based upon second-order conditional independence in order to facilitate posterior updating and prediction of required distributional quantities. Such models are formulated particularly for multivariate regression and time series analysis with unknown observational variance–covariance components. The similarities and differences of these models with the Bayes linear approach are established. Several subclasses of important models, including regression and time series models with errors following multivariate t, inverted multivariate t and Wishart distributions, are discussed in detail. Two numerical examples consisting of simulated data and of US investment and change in inventory data illustrate the proposed methodology.

Bite force is an ecologically important biomechanical performance measure that is informative in inferring the ecology of extinct taxa. However, biomechanical modelling to estimate bite force is associated with some level of uncertainty. Here, I assess the accuracy of bite force estimates in extinct taxa using a Bayesian phylogenetic prediction model. I first fitted a phylogenetic regression model on a training set comprising extant data. The model predicts bite force from body mass and skull width while accounting for differences owing to biting position. The posterior predictive model has a 93% prediction accuracy as evaluated using leave‐one‐out cross‐validation. I then predicted bite force in 37 species of extinct mammals and archosaurs from the posterior distribution of predictive models, generating posterior predictive distributions of null expectations given body mass, skull width and phylogenetic position. Biomechanically estimated bite forces from the literature fall within the posterior predictive distributions for all except four species of extinct taxa and are thus as accurate as predicted from body size and skull width, given the variation inherent in extant taxa and the amount of time available for variance to accrue. Biomechanical modelling remains a valuable means to estimate bite force in extinct taxa and should be reliably informative of functional performances and serve to provide insights into past ecologies.

Statistical parametric methodologies are widely employed in the analysis of time series of gas turbine sensor readings. These methodologies identify outliers as a consequence of excessive deviation from a statistically-based model, derived from available observations. Among parametric techniques, the k-σ methodology demonstrates its effectiveness in the analysis of stationary time series. Furthermore, the simplicity and the clarity of this approach justify its direct application to industry. On the other hand, the k-σ methodology usually proves to be unable to adapt to dynamic time series, since it identifies observations in a transient as outliers. As this limitation is caused by the nature of the methodology itself, two improved approaches are considered in this paper in addition to the standard k-σ methodology. The two proposed methodologies maintain the same rejection rule of the standard k-σ methodology, but differ in the portions of the time series from which statistical parameters (mean and standard deviation) are inferred. The first approach performs statistical inference by considering all observations prior to the current one, which are assumed reliable, plus a forward window containing a specified number of future observations. The second approach proposed in this paper is based on a moving window scheme. Simulated data are used to tune the parameters of the proposed improved methodologies and to prove their effectiveness in adapting to dynamic time series. The moving window approach is found to be the best on simulated data in terms of True Positive Rate (TPR), False Negative Rate (FNR) and False Positive Rate (FPR). Therefore, the performance of the moving window approach is further assessed towards both different simulated scenarios and field data taken on a gas turbine.

Chapter 5 - Fitting Models Using the Bayesian Modeling Software BUGS and JAGS

In this paper we use the Gibbs sampling algorithm to present a Bayesian analysis to multiplicative double seasonal autoregressive (DSAR) models, considering both estimation and prediction problems. Assuming the model errors are normally distributed and using natural conjugate and g priors on the initial values and model parameters, we show that the conditional posterior distributions of the model parameters and variance are multivariate normal and inverse gamma respectively, and the conditional predictive distribution of the future observations is a multivariate normal. Using these closed-form conditional posterior and predictive distributions, we apply the Gibbs sampling to approximate empirically the marginal posterior and predictive distributions, enabling us easily to carry out multiple-step ahead predictions. The proposed Bayesian method is evaluated using simulation study and real-world time series dataset.

In modeling HIV/AIDS progression, we carried out a comprehensive investigation into the risk factors for statespecific-failure rates to identify the influential co-variates using Bayesian Model averaging method (BMA).BMA provides a posterior probability via Markov Chain Monte Carlo (MCMC) for each variable that belongs to the model.It accounts for model uncertainty by averaging all plausible models using their posterior probabilities as the weights for model-averaged predictions and estimates of the required parameters.Patients' age, and gender, among other co-variates, have been found to influence the state-specific-failure rates highly.However, the impact of each of the factors on the state specific-failure was not quantified.This paper seeks to evaluate and quantify the contribution of the patient's age and gender, CD4 cell count during any two consecutive visits, and state movement on the state-specific-failure rates for patients transiting either to the same, better or worse state.We used R Studio statistical Programming software to implement the method by applying BMS and BMA packages.State movement had a comparatively large coefficient with a posterior inclusion probability (PIP) of 0.8788 (87.88%).Hence, the most critical variable followed by observation-two-CD4-cell-count with a PIP of 0.1416 (14.16%), age and gender were the last with a PIP of 0.0556 (5.56%) and 0.0510 (5.10%) respectively for patients transiting to the same state.For patients transiting to a better state, the patients' age group dominated with a PIP of 0.9969 (99.69%), followed by patients' gender with a PIP of 0.0608 (6.08%).Patients' CD4 cell count during the second observation had the least PIP of 0.0399 (3.99%).For patients transiting to a worse disease state, patients CD4 cell count during the second observation proved to be the most important, with a PIP of 0.6179(61.79%)followed by state movement with a PIP of 0.2599 (25.99%), patients gender tailed with a PIP of 0.0467 (4.67%).

Objective: Serially measure the rate of hematoma volume (HV) and edema volume (EV) resolution in patients with acute-to-early chronic ICH Background: After ICH, the primary cause of brain injury is the mechanical force from the compression of the brain tissue. The secondary damage involves (bio)chemical injury produced by the toxicity of components of extravasated blood, which directly compromises surrounding tissue by oxidative stress, cell death, neuroinflammation, and brain edema. There is very little known about the rate of HV and EV resolution within one month of ictus. Methods: Twelves patients, a placebo sub-cohort of a previous clinical trial, were serially imaged on day-1, 14, and 28 of initial ictus. Anatomical (FLAIR, T1W, and T2W) images were obtained on a 3T MRI system. HV and EV were segmented using semiautomated seed growing algorithm. The prediction of HV and EV resolution rate was calculated using the Bayesian model and compared across the different time points. Results: There were 7M/5F with the average age of 57 ± 9y. The baseline average HV/NIHSS 22,998 mm 3 /11. All patients with deep ICH and 6 extended to IVH. The multilevel Bayesian analysis showed the rate of HV resolution -327.90 mm 3 (95% CrI = -475.37- -181.86; Posterior Probability b < 0 ≥ 0.99) with 14 days acceleration of -17.67 mm 3 (95% CrI = -35.89- 1.21; Posterior Probability b < 0 = 0.97). The rate of EV resolution -341.44 mm 3 (95% CrI = -736.32- 60.98; Posterior Probability b < 0 = 0.96) with 14 days acceleration -34.63 mm 3 (95% CrI = -85.47- 16.61; Posterior Probability b < 0 = 0.91). Overall the decrease in HV over time was an offsetting deceleration which increases over time. The overall trend of HV resolution is shown in Fig-1 Conclusion: There was a high variance in the rate of HV and EV resolution, but overall all the patients exhibited a similar trend. The rates of HV and EV resolution can help us provide optimal patient care.