Articles

16.1: Dynamical Network Models - Mathematics


There are several different classes of dynamical network models. In this chapter, we will discuss the following three, in this particular order:

  • Models for “dynamics on networks” These models are the most natural extension of traditional dynamical systems models. They consider how the states of components, or nodes, change over time through their interactions with other nodes that are connected to them. The connections are represented by links of a network, where the network topology is fixed throughout time. Cellular automata, Boolean networks, and artificial neural networks (without learning) all belong to this class.
  • Models for “dynamics of networks” These are the models that consider dynamical changes of network topology itself over time, for various purposes: to understand mechanisms that bring particular network topologies, to evaluate robustness and vulnerability of networks, to design procedures for improving certain properties of networks, etc. The dynamics of networks are a particularly hot topic in network science nowadays (as of 2015) because of the increasing availability of temporal network data [65].
  • Models for “adaptive networks” I must admit that I am not 100% objective when it comes to this class of models, because I am one of the researchers who have been actively promoting it [66]. Anyway, the adaptive network models are models that describe the co-evolution of dynamics on and of networks, where node states and network topologies dynamically change adaptively to each other. Adaptive network models try to unify different dynamical network models to provide a generalized modeling framework for complex systems, since many real-world systems show such adaptive network behaviors [67].

A Dynamic Network Model of Interbank Lending—Systemic Risk and Liquidity Provisioning

We develop a dynamic model of interbank borrowing and lending activities in which banks are organized into clusters, and adjust their monetary reserve levels to meet prescribed capital requirements. Each bank has its own initial monetary reserve level and faces idiosyncratic risks characterized by an independent Brownian motion, whereas system wide, the banks form a hierarchical structure of clusters. We model the interbank transactional dynamics through a set of interacting measure-valued processes. Each individual process describes the intracluster borrowing/lending activities, and the interactions among the processes capture the intercluster financial transactions. We establish the weak limit of the interacting measure-valued processes as the number of banks in the system grows large. We then use the weak limit to develop asymptotic approximations of two proposed macromeasures (the liquidity stress index and the concentration index), both capturing the dynamics of systemic risk. We use numerical examples to illustrate the applications of the asymptotics and conduct-related sensitivity analysis with respect to various indicators of financial activity.


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Dynamic systems in simple terms, are systems that evolve over time. What evolves over time? The states of the system. What are states? States are variables that describe the system. For instance, in the case of motion of celestial bodies, we could consider the current position and velocity as the states. These states can be used to predict the future position of the celestial body. In this example, the motion of the celestial body is the dynamic system.

More formally, any sys t em in which the future states are dependent on the current state along with a set of evolution rules is called a dynamic system. Going back to the celestial body example, we know that its motion is bound by the laws of gravitation.

In order to better understand these systems and make reasonable predictions, we rely on mathematical equations to represent them. In the case of dynamic systems, we use ordinary differential equations (ODEs). Simply put, ODEs are equations relating functions and their derivatives. Particularly, in dynamic systems we deal with derivatives with respect to time. A general form of a single variable dynamic system can be represented as an ODE shown below.

There are several numerical methods available to solve ODEs like Euler’s method, implicit Euler’s method, Runge-Kutta methods, and more. So why do we need another method? Why do we need deep learning? While it is true that several numerical approximations exist for solving ODEs, in practice however, it is quite complicated to obtain these equations for highly nonlinear systems. This is because it is difficult to account for all the factors that affect the system. For example, in the case of weather predictions, it is simply impossible to even know all the variables affecting them. We therefore rely on data to model these systems.

In a first-principles model, a set of mathematical equations obtained from the physical laws of conservation like mass balance, energy balance, and momentum balance are used to represent the system. Whereas, in data-based models (black-box models) we try to learn the behaviour of the process from the data collected. As shown below, we do not know the true underlying phenomena. We therefore use the input and output data to identify a model that best represents the system.

In this article, we are going to be focusing on long short-term memory (LSTM) networks to model nonlinear dynamic systems. A basic understanding of recurrent neural networks (RNN) and LSTM would be helpful. To better understand the workings of RNN and LSTM, refer to these beautifully illustrative articles shown below.


16.1: Dynamical Network Models - Mathematics

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John Conway, one of the most remarkable mathematicians of our time, died on April 11, 2020, at the age of 82. His interests were so varied and his contributions so numerous that it is difficult to characterize them in a few words. To begin to appreciate his legacy see this obituary from the New York Times and this article from Princeton University. For more mathematical details about his accomplishments see this tribute from Ed Dunne.


About the Guest Editors:

Konstantin Blyuss

Guest Editor, PLOS ONE, PLOS Biology, and PLOS Computational Biology

Konstantin Blyuss is a Reader in the Department of Mathematics at the University of Sussex, UK. He obtained his PhD in applied mathematics at the University of Surrey, which was followed by PostDocs at Universities of Exeter and Oxford. Before coming to Sussex in 2010, he was a Lecturer in Complexity at the University of Bristol. His main research interests are in the area of dynamical systems applied to biology, with particular interest in modelling various aspects of epidemiology, dynamics of immune responses and autoimmunity, as well as understanding mechanisms of interactions between plants and their pathogens

Sara del Valle

Guest Editor, PLOS ONE, PLOS Biology, and PLOS Computational Biology

Dr. Sara Del Valle is a scientist and deputy group leader in the Information Systems and Modeling Group at Los Alamos National Laboratory. She earned her Ph.D. in Applied Mathematics and Computational Science in 2005 from the University of Iowa. She works on developing, integrating, and analyzing mathematical, computational, and statistical models for the spread of infectious diseases such as smallpox, anthrax, HIV, influenza, malaria, Zika, Chikungunya, dengue, and Ebola. Most recently, she has been investigating the role of heterogeneous data streams such as satellite imagery, Internet data, and climate on detecting, monitoring, and forecasting diseases around the globe. Her research has generated new insights on the impact of behavioral changes on diseases spread as well as the role of non-traditional data streams on disease forecasting.

Jennifer Flegg

Guest Editor, PLOS ONE, PLOS Biology, and PLOS Computational Biology

Jennifer Flegg is a Senior Lecturer and DECRA fellow in the School of Mathematics and Statistics at the University of Melbourne. Her research focuses on mathematical biology in areas such as wound healing, tumour growth and epidemiology. She was awarded a PhD in 2009 from Queensland University of Technology on mathematical modelling of tissue repair. From 2010 &ndash 2013, she was at the University of Oxford developing statistical models for the spread of resistance to antimalarial drugs. From 2014 &ndash April 2017 she was a Lecturer in the School of Mathematical Sciences at Monash University. In May 2017 she joined the School of Mathematics and Statistics at the University of Melbourne as a Senior Lecturer in Applied Mathematics.

Louise Matthews

Guest Editor, PLOS ONE, PLOS Biology, and PLOS Computational Biology

Louise Matthews is Professor of Mathematical Biology and Infectious Disease Ecology at the Institute of Biodiversity, Animal Health and Comparative Medicine (BAHCM) at the University of Glasgow. She holds a degree and PhD in mathematics and has over 20 years research experience as an epidemiologist, with a particular focus on diseases of veterinary and zoonotic importance. Her current interests include a focus on drug resistance antibiotic resistance in livestock the community and the healthcare setting anthelminthic resistance in livestock and drug resistance in African Animal Trypanosomiasis. She is also interested in the integration of economic and epidemiological approaches such as game theory to understand farmer behaviour and micro-costing approaches to promote adoption of measures to reduce antibiotic resistance.

Jane Heffernan

Guest Editor, PLOS ONE, PLOS Biology, and PLOS Computational Biology

Jane Heffernan is a Professor in the Department of Mathematics and Statistics at York University, and York Research Chair (Tier II). She is also the Director of the Centre for Disease Modelling (CDM), and serves on the Board of Directors of the Canadian Applied and Industrial Mathematics Society (CAIMS). She is also very active in the Society for Mathematical Biology (SMB). Dr. Heffernan&rsquos research program centers on understanding the spread and persistence of infectious diseases. Her Modelling Infection and Immunity Lab focuses on the development of new biologically motivated models of infectious diseases (deterministic and stochastic) that describe pathogen dynamics in-host (mathematical immunology) and in a population of hosts (mathematical epidemiology), as well as models in immuno-epidemiology, which integrate the in-host dynamics with population level models. More recently, Heffernan is focusing on applying mathematics and modelling to studying pollinator health and disease biology.


Discussion

In this work, we have explored the importance of developing detailed dynamic models able to support accurate phenotype predictions and their use in efficient strain optimization algorithms, a field that has the potential to produce significant impact in industrial biotechnology. Methods to fulfill these tasks and to generate new knowledge are emerging and being evaluated, with the intention of exploring and bridging the gap between different (bio)mathematical modeling frameworks. Systems biology provides strategies for ME to take advantage of the best simulation and optimization method features, and to deal with the most remarkable limitation regarding the lack of available experimental information, which affects accuracy and feasibility of solutions (Machado et al., 2012). Another identified issue is the addition of more detailed information to already existing genome-scale models, to increase their scalability and the range of industrial applications for strain design. The global challenge consists in generating high quality models that make a difference in the improvement of performance of CSOMs for larger scales. However, this task becomes a challenge since metabolic simulation has to deal with thousands of reaction rates and metabolite concentrations.

Moreover, current research trends point to the inclusion of parameter uncertainty to increase the level of flexibility, using kinetic models built with stochastic optimization methods, and also improvement of phenotype predictions by using complementary data, such as various types of omics data, particularly gene expressions (Jahan et al., 2016). Furthermore, some methodologies include evaluation of stability, robustness and other type of analyses of dynamic features, such as oscillations (Schaefer et al., 1999). Additionally, identifiability analysis helps to drive dynamic models to the development of better experimental techniques and to polish methods to solve the optimization problems, focusing on exact or stochastic formulations (Villaverde and Barreiro, 2016).

The revised dynamic modeling approaches are supported by the use of optimization methods for two main identified tasks. On one hand, we have the development of models for phenotype prediction, particularly for parameter estimation. This task requires model calibration through the minimization of differences between predicted and experimental values (Banga, 2008). On the other hand, we have the strain design task, which aims at finding the optimal interventions strategies for producing strains with enhanced capabilities (Vital-Lopez et al., 2006). Both tasks have to deal with choosing the most suitable optimization method, which depends on the type of problem or application. However, it has been noticed that stochastic optimization approaches seem to be an acceptable option to avoid issues with scalability, flexibility or convergence time. Involving metaheuristics, the search of alternatives in bigger solution spaces is more efficient for complex (multi)objective functions. For example, finding the optimal kinetic parameters to reach desired phenotypes in a genome scale, once a suitable known model structure is identified, such as central carbon metabolism of E. coli or S. cerevisiae (Rocha et al., 2008).

Kinetic models can also be studied within hybrid models for phenotype prediction, which allows the integration of available kinetic relations with genome-scale constraint-based model formalisms. This idea results in more detailed descriptions of large-scale models, since the effect of metabolite concentrations and substrate-level enzyme regulation cannot be captured with stoichiometry-only metabolic models and analysis methods (Chowdhury et al., 2014). These models can be used as a base for optimization methods for strain design to identify genetic perturbations that are consistent with enzyme expressions and metabolite concentrations. In principle, the algorithms can perform the combination of CSOMs that use only constraint-based or only kinetics-based models. The integration of approaches has become very promising to accelerate the process of innovation in the world of ME, leading to the targeted overproduction of desired chemicals.

The mentioned CSOMs using hybrid models, such as the one used by Chowdhury and coworkers, attempt to identify a minimal set of interventions on enzymatic parameter changes and reaction flux changes, such that less rearrangements of the flux distribution are required, and concentration bounds are not violated (Chowdhury et al., 2014). An important remark is that CSOMs with hybrid approaches can always be improved incorporating available omics information, to sharpen the prediction fidelity. More constraints can be added to the optimization problems to restrict more the flux ranges and to decrease the space of possible solutions. Also, temporal consideration can be addressed by integrating hybrid CSOMs with the DFBA framework (Mahadevan et al., 2002) to explore the variation of metabolic modifications as a function of time suggesting similar actions of ribonucleic acid interference type of interventions.

We emphasize as one of the major outcomes from the revision of phenotype prediction and strain optimization methods, the fact that the selection of a specific dynamic modeling approach is always subject to the prospective application, as well as the amount and type of experimental data available. While and adequate path is difficult to define for all cases, Table 1 can provide an aid in this process. For instance, mechanistic dynamic modeling has been widely used to have conventional expressions describing the structural features of metabolic systems. However, this represents a general approach that might not be able to detail specific biological processes, for which other methods can be used. One example of this is the use of log-linear approximations to accurately describe dynamic responses of poorly known non-linear systems, however, parameter estimation methods might be limited in their ability to satisfactorily fit data from observations when small concentration of metabolites are present. This kind of problem can be tackled using convenience rate laws, since they require a small number of parameters that can be easily computed. Additionally, cooperativity and saturation expressions are used to fit experimental data for systems with a saturable form, while modular rate laws can simplify thermodynamic-kinetic modeling formalisms. By providing analytical solutions and avoiding the use of non-linear problems, the computational burden and convergence times are greatly reduced.

Furthermore, stochastic approaches are able to capture variability in the described species, for very well-known structural features, such as stiff systems. The dynamics are simulated by knowing the probabilities of transitions between every possible state. However, the formulation can become a non-trivial stochastic simulation algorithm for networks when the number of reacting molecules is large per reactant, which is common for most of realistic kinetic models. The aim will be always to establish a trade-off between the size of the model and the level of accuracy of the solutions, depending on the type of experimental data, which is not always available in the literature. Finally, efforts for enforcing the inclusion of experimental data as Supplementary Material when publishing new findings should be taken by publishers, a measure that would greatly improve the development of these type of methods.

Also, studying kinetic models, especially used within hybrid models, has revealed strengths and limitations of model-driven strain design, and indicated that kinetic models have the potential to substantially over-perform FBA-based predictions when parameterized under similar conditions, but may perform worse than FBA when predicting a significantly different metabolic phenotype. Studies have also demonstrated the need to perform model parameterization for a diverse range of genetic or environmental perturbations, and the tight integration of transcriptional level along with substrate-level regulatory interactions. At a fundamental level, kinetic models must be a priori provided with the quantitative description and as many as possible regulatory switches in response to genetic or environmental perturbations. The quality of mechanistic information enables a detailed description of metabolism such as dynamics, enzyme activities, and metabolite concentrations but can result in erroneous predictions since some modeling assumptions can be missing or incorrect. Nevertheless, by studying failure modes of kinetic models, valuable information can be uncovered for restoring prediction consistency for new phenotypes (Khodayari et al., 2015). These findings, together with reported experimental evaluations of constraint-based CSOMs, available in the literature since the last two decades (Maia et al., 2016), help building the case for the combination of these approaches with kinetics-based models, becoming tools for the optimization of bioprocesses for a wide range of industrially relevant chemicals.

Additional studies, not covered in detail in this review, point to the use of artificial intelligence as a different approach to analyze mathematical models for ME purposes. Novel technologies applied to metabolomics can substantially improve search algorithms to increase the dynamic range, number of carbon-carrying metabolites and possible pathways to transform a given source metabolite into a given target metabolite (Kell, 2006). One example of the use of artificial intelligence to accelerate the design of microbial cell factories, is the development of an efficient workflow for combinatorial optimization of the large biosynthetic genotypic space of heterologous metabolic pathways in yeast. This method is able to precisely tune the expression level of genes with a machine learning algorithm based on an artificial neural network ensemble to avoid over-fitting, and it is also able to predict strains with titer improvements among several possible designs (Zhou et al., 2018). Another recent advance on exploiting artificial intelligence techniques is an approach that combines machine learning and abundant multiomics data (proteomics and metabolomics) to effectively predict pathway dynamics. The method outperforms a classical Michaelis–Menten kinetic model, and produces qualitative and quantitative predictions that can be used to productively guide bioengineering efforts, by using only two time-series as training data. This work shows that, given sufficient data, the dynamics of complex coupled non-linear systems can be systematically learned (Costello and Martin, 2018). Finally, the development of biological models based on artificial intelligence has been analyzed in a recent review, which highlights the scope of information collections, database constructions, and machine learning techniques that can facilitate strain design (Oyetunde et al., 2018).

In this review, we analyzed the main mathematical formalisms used for dynamic/hybrid modeling of microbial metabolism, scrutinized their inclusion into strain optimization applications and made a critical evaluation of future steps in this research topic. The ladder toward more realistic strain engineering strategies seems to be undeniably limited by more detailed representation of the dynamics of the biological systems. However, as was thoroughly discussed in this work, these are still hindered by the lack of appropriate parameters even for the most studied organisms such as E. coli and S. cerevisiae. Thus, the advancement of the field will always be dependent on incremental investments in fundamental research geared toward the deeper understanding of biological mechanisms, including local phenomena. For this purpose, the use of hybrid models as flexible vessels for the cumulative inclusion of knowledge will be of major importance in the coming years.


Introduction to Econometrics with R

A Vector autoregressive (VAR) model is useful when one is interested in predicting multiple time series variables using a single model. At its core, the VAR model is an extension of the univariate autoregressive model we have dealt with in Chapters 14 and 15. Key Concept 16.1 summarizes the essentials of VAR.

Key Concept 16.1

Vector Autoregressions

The vector autoregression (VAR) model extends the idea of univariate autoregression to (k) time series regressions, where the lagged values of all (k) series appear as regressors. Put differently, in a VAR model we regress a vector of time series variables on lagged vectors of these variables. As for AR( (p) ) models, the lag order is denoted by (p) so the VAR( (p) ) model of two variables (X_t) and (Y_t) ( (k=2) ) is given by the equations

[egin Y_t =& , eta_ <10>+ eta_ <11>Y_ + dots + eta_ <1p>Y_ + gamma_ <11>X_ + dots + gamma_ <1p>X_ + u_<1t>, X_t =& , eta_ <20>+ eta_ <21>Y_ + dots + eta_ <2p>Y_ + gamma_ <21>X_ + dots + gamma_ <2p>X_ + u_<2t>. end]

The (eta) s and (gamma) s can be estimated using OLS on each equation. The assumptions for VARs are the time series assumptions presented in Key Concept 14.6 applied to each of the equations.

It is straightforward to estimate VAR models in R. A feasible approach is to simply use lm() for estimation of the individual equations. Furthermore, the Rpackage vars provides standard tools for estimation, diagnostic testing and prediction using this type of models.

When the assumptions of Key Concept 16.1 hold, the OLS estimators of the VAR coefficients are consistent and jointly normal in large samples so that the usual inferential methods such as confidence intervals and (t) -statistics can be used.

The structure of VARs also allows to jointly test restrictions across multiple equations. For instance, it may be of interest to test whether the coefficients on all regressors of the lag (p) are zero. This corresponds to testing the null that the lag order (p-1) is correct. Large sample joint normality of the coefficient estimates is convenient because it implies that we may simply use an (F) -test for this testing problem. The explicit formula for such a test statistic is rather complicated but fortunately such computations are easily done using the R functions we work with in this chapter. Another way to determine optimal lag lengths are information criteria like the (BIC) which we have introduced for univariate time series regressions in Chapter 14.6. Just as in the case of a single equation, for a multiple equation model we choose the specification which has the smallest (BIC(p)) , where [egin BIC(p) =& , logleft[ ext(widehat_u) ight] + k(kp+1) frac. end] with (widehat_u) denoting the estimate of the (k imes k) covariance matrix of the VAR errors and ( ext(cdot)) denotes the determinant.

As for univariate distributed lag models, one should think carefully about variables to include in a VAR, as adding unrelated variables reduces the forecast accuracy by increasing the estimation error. This is particularly important because the number of parameters to be estimated grows qudratically to the number of variables modeled by the VAR. In the application below we shall see that economic theory and empirical evidence are helpful for this decision.

A VAR Model of the Growth Rate of GDP and the Term Spread

We now show how to estimate a VAR model of the GDP growth rate, (GDPGR) , and the term spread, (TSpread) . As following the discussion on nonstationarity of GDP growth in Chapter 14.7 (recall the possible break in the early 1980s detected by the (QLR) test statistic), we use data from 1981:Q1 to 2012:Q4. The two model equations are

[egin GDPGR_t =& , eta_ <10>+ eta_ <11>GDPGR_ + eta_ <12>GDPGR_ + gamma_ <11>TSpread_ + gamma_ <12>TSpread_ + u_<1t>, TSpread_t =& , eta_ <20>+ eta_ <21>GDPGR_ + eta_ <22>GDPGR_ + gamma_ <21>TSpread_ + gamma_ <22>TSpread_ + u_<2t>. end]

The data set us_macro_quarterly.xlsx is provided on the companion website to Stock and Watson (2015) and can be downloaded here. It contains quarterly data on U.S. real (i.e., inflation adjusted) GDP from 1947 to 2004. We begin by importing the data set and do some formatting (we already worked with this data set in Chapter 14 so you may skip these steps if you have already loaded the data in your working environment).

We estimate both equations separately by OLS and use coeftest() to obtain robust standard errors.

We end up with the following results:

The function VAR() can be used to obtain the same coefficient estimates as presented above since it applies OLS per equation, too.

VAR() returns a list of lm objects which can be passed to the usual functions, for example summary() and so it is straightforward to obtain model statistics for the individual equations.

We may use the individual model objects to conduct Granger causality tests.

Both Granger causality tests reject at the level of (5\%) . This is evidence in favor of the conjecture that the term spread has power in explaining GDP growth and vice versa.

Iterated Multivariate Forecasts using an Iterated VAR

The idea of an iterated forecast for period (T+2) based on observations up to period (T) is to use the one-period-ahead forecast as an intermediate step. That is, the forecast for period (T+1) is used as an observation when predicting the level of a series for period (T+2) . This can be generalized to a (h) -period-ahead forecast where all intervening periods between (T) and (T+h) must be forecasted as they are used as observations in the process (see Chapter 16.2 of the book for a more detailed argument on this concept). Iterated multiperiod forecasts are summarized in Key Concept 16.2.

Key Concept 16.2

Iterated Multiperiod Forecasts

The steps for an iterated multiperiod AR forecast are:

Estimate the AR( (p) ) model using OLS and compute the one-period-ahead forecast.

Use the one-period-ahead forecast to obtain the two-period-ahead forecast.

Continue by iterating to obtain forecasts farther into the future.

An iterated multiperiod VAR forecast is done as follows:

Estimate the VAR( (p) ) model using OLS per equation and compute the one-period-ahead forecast for all variables in the VAR.

Use the one-period-ahead forecasts to obtain the two-period-ahead forecasts.

Continue by iterating to obtain forecasts of all variables in the VAR farther into the future.

Since a VAR models all variables using lags of the respective other variables, we need to compute forecasts for all variables. It can be cumbersome to do so when the VAR is large but fortunately there are R functions that facilitate this. For example, the function predict() can be used to obtain iterated multivariate forecasts for VAR models estimated by the function VAR().

The following code chunk shows how to compute iterated forecasts for GDP growth and the term spread up to period 2015:Q1, that is (h=10) , using the model object VAR_est.

This reveals that the two-quarter-ahead forecast of GDP growth in 2013:Q2 using data through 2012:Q4 is (1.69) . For the same period, the iterated VAR forecast for the term spread is (1.88) .

The matrices returned by predict(VAR_est) also include (95\%) prediction intervals (however, the function does not adjust for autocorrelation or heteroskedasticity of the errors!).

We may also plot the iterated forecasts for both variables by calling plot() on the output of predict(VAR_est) .

Direct Multiperiod Forecasts

A direct multiperiod forecast uses a model where the predictors are lagged appropriately such that the available observations can be used directly to do the forecast. The idea of direct multiperiod forecasting is summarized in Key Concept 16.3.

Key Concept 16.3

Direct Multiperiod Forecasts

A direct multiperiod forecast that forecasts (h) periods into the future using a model of (Y_t) and an additional predictor (X_t) with (p) lags is done by first estimating

[egin Y_t =& , delta_0 + delta_1 Y_ + dots + delta_

Y_ + delta_ X_ +& dots + delta_ <2p>Y_ + u_t, end]

which is then used to compute the forecast of (Y_) based on observations through period (T) .

For example, to obtain two-quarter-ahead forecasts of GDP growth and the term spread we first estimate the equations

[egin GDPGR_t =& , eta_ <10>+ eta_ <11>GDPGR_ + eta_ <12>GDPGR_ + gamma_ <11>TSpread_ + gamma_ <12>TSpread_ + u_<1t>, TSpread_t =& , eta_ <20>+ eta_ <21>GDPGR_ + eta_ <22>GDPGR_ + gamma_ <21>TSpread_ + gamma_ <22>TSpread_ + u_ <2t>end]

and then substitute the values of (GDPGR_<2012:Q4>) , (GDPGR_<2012:Q3>) , (TSpread_<2012:Q4>) and (TSpread_<2012:Q3>) into both equations. This is easily done manually.

Applied economists often use the iterated method since this forecasts are more reliable in terms of (MSFE) , provided that the one-period-ahead model is correctly specified. If this is not the case, for example because one equation in a VAR is believed to be misspecified, it can be beneficial to use direct forecasts since the iterated method will then be biased and thus have a higher (MSFE) than the direct method. See Chapter 16.2 for a more detailed discussion on advantages and disadvantages of both methods.

References

Stock, J. H., and M. W. Watson. 2015. Introduction to Econometrics, Third Update, Global Edition. Pearson Education Limited.


Results for a Circular Road

A circular road is a particular friendly case for an analysis, since the total number of vehicles is exactly conserved. If the road is not too long, traffic will in general form one single traveling wave, i.e. a single jamiton, and thus a single shock is observed. Below figures and videos show the results of simulations and theoretical predictions for the case of a circular road of length 230m.

In the case of inviscid equations a sharp shock is realized. Here, the final solution is predicted theoretically. The match between theory and numerical results is generally very good. While the inviscid equations allow a simple analysis, using the Rankine-Hugoniot conditions at the shock, the resulting vehicle behavior is somewhat extreme. Vehicles slow down from high to low velocity in zero time.


Watch the video: Μη Γραμμικά Δυναμικά (October 2021).