Computation and Modeling for Fractional Order Systems

By Snehashish Chakraverty

Computation and Modeling for Fractional Order Systems provides readers with problem-solving techniques for obtaining exact and/or approximate solutions of governing equations arising in fractional dynamical systems presented using various analytical, semi-analytical, and numerical methods. In this regard, this book brings together contemporary and computationally efficient methods for investigating real-world fractional order systems in one volume. Fractional calculus has gained increasing popularity and relevance over the last few decades, due to its well-established applications in various fields of science and engineering. It deals with the differential and integral operators with non-integral powers. Fractional differential equations are the pillar of various systems occurring in a wide range of science and engineering disciplines, namely physics, chemical engineering, mathematical biology, financial mathematics, structural mechanics, control theory, circuit analysis, and biomechanics, among others. The fractional derivative has also been used in various other physical problems, such as frequency-dependent damping behavior of structures, motion of a plate in a Newtonian fluid, PID controller for the control of dynamical systems, and many others. The mathematical models in electromagnetics, rheology, viscoelasticity, electrochemistry, control theory, Brownian motion, signal and image processing, fluid dynamics, financial mathematics, and material science are well defined by fractional-order differential equations. Generally, these physical models are demonstrated either by ordinary or partial differential equations. However, modeling these problems by fractional differential equations, on the other hand, can make the physics of the systems more feasible and practical in some cases. In order to know the behavior of these systems, we need to study the solutions of the governing fractional models. The exact solution of fractional differential equations may not always be possible using known classical methods. Generally, the physical models occurring in nature comprise complex phenomena, and it is sometimes challenging to obtain the solution (both analytical and numerical) of nonlinear differential equations of fractional order. Various aspects of mathematical modeling that may include deterministic or uncertain (viz. fuzzy or interval or stochastic) scenarios along with fractional order (singular/non-singular kernels) are important to understand the dynamical systems. Computation and Modeling for Fractional Order Systems covers various types of fractional order models in deterministic and non-deterministic scenarios. Various analytical/semi-analytical/numerical methods are applied for solving real-life fractional order problems. The comprehensive descriptions of different recently developed fractional singular, non-singular, fractal-fractional, and discrete fractional operators, along with computationally efficient methods, are included for the reader to understand how these may be applied to real-world systems, and a wide variety of dynamical systems such as deterministic, stochastic, continuous, and discrete are addressed by the authors of the book.

• Language: English
• Publisher: Academic Press
• Release date: Feb 20, 2024
• ISBN: 9780443154058

Book preview

1: Response time and accuracy modeling through the lens of fractional dynamics

A foundational primer

Amir Hosein Hadian Rasanana; Nathan J. Evansb,c; Jörg Rieskampa; Jamal Amani Radd    aDepartment of Psychology, University of Basel, Basel, Switzerland

bDepartment of Psychology, Ludwig Maximilian University of Munich, Munich, Germany

cSchool of Psychology, The University of Queensland, St Lucia, QLD, Australia

dInstitute for Cognitive and Brain Sciences, Shahid Beheshti University, Tehran, Iran

Abstract

Decision making is a cognitive process that is both an important aspect of human cognition and fundamental to understanding other basic cognitive functions, such as attention, memory, and learning. By exploring the underlying mechanism of decision making, we can obtain better insight into other cognitive processes. Response time and accuracy are two important sources of behavioral data that can shed light on the cognitive process of decision making. Among the cognitive decision-making models that can account for both response time and accuracy data, diffusion models, which belong to the class of sequential sampling models, have proven to be very successful in predicting behavior in various domains. Although diffusion models are very popular, they have some limitations (e.g., capturing fast errors or generating Gaussian-like response time distributions). Therefore, an extension of the diffusion model named the Lévy flight model was successfully introduced and could show improved prediction accuracy. However, the Lévy model also had some limitations, including the lack of a mathematically tractable likelihood function and the ability to model non-binary decisions. Furthermore, the model generates a non-realistic accumulation path when the Lévy index tends to a value of 1. In this chapter, we first review the basics of sequential sampling models and present one of the most prominent sequential sampling models, the drift-diffusion model. We then discuss Lévy flight models and their limitations. Thereafter, we introduce the Lévy–Brownian model representing the general form of a diffusion model. Next, the estimation procedure of this model is presented using a fractional partial differential equation, and it will be fitted on a sample behavioral dataset. We close the book chapter by summarizing the significant advancement in cognition by following the suggested models.

Keywords

sequential sampling models; decision making; Lévy–Brownian process; fractional calculus

1.1 Introduction

Researchers have attempted to explain the underlying mechanisms of the human decision-making process for many decades. This general fascination led to the proposal of various theoretical and computational frameworks for understanding the decision-making process, with theories being developed by researchers from different fields of science, including neuroscience, economics, psychology, and cognitive science. Among all of these attempts, some theoretical frameworks – such as cumulative prospect theory, utility theory, signal detection theory, general recognition theory, or sequential sampling theory – have attracted much scientific attention. Each of these frameworks shares the common property of being mathematical models of human decision making, which attempt to formalize the theoretical beliefs of researchers into an exact mathematical functional form, which can then be directly fit to empirical data to assess consistency and provide estimates of theoretically relevant constructs. However, between computational theories of decision making, which have the advantage of being able to account for both response times and choice behavior, sequential sampling models (SSMs) have become more popular.

SSMs assume that the decision maker samples and accumulates information in favor of each decision alternative. The decision is triggered whenever accumulated information reaches a specific threshold for one alternative. Basically, SSMs can differ based on how evidence is accumulated. Evidence accumulation can be done in discrete (e.g., accumulator model [1]) (equidistant or random) time intervals or continuously through time (e.g., diffusion decision model [2]). On the other hand, the amount of information may be considered as fixed-sized chunks (e.g., recruitment model [3]) or chunks of varying sizes (e.g., Poisson counter model [4]) [5,6]. Recent work introduced some new concepts for SSMs so that SSMs can also be categorized according to the type of evidence that is accumulated and the type of options considered (Fig. 1.1).

Figure 1.1 Categorization of different sequential sampling models.

From the mathematical point of view, the accumulation process of an SSM could be considered a Brownian motion with the following formulation:

(1.1)

where is the accumulator and shows the amount of accumulated evidence unit time t, is the rate of accumulation and is named drift rate, is the diffusion coefficient, and is the noise term and represents the inter-trial variability. This formula is the general form of the diffusion model. Basically, the accumulation process starts at some points within an interval surrounded by absorbing [2,7] or reflecting boundaries [8,9].

1.1.1 Historical foundation and applications of sequential sampling theory

The historical foundation of SSMs began in 1960 when Stone [10] introduced the "likelihood ratio model as a random walk model for presenting the process of information accumulation. This work was the first research in psychology in which a random walk model based on the sequential sampling ratio test [11] was considered for modeling choice response time. After that, Stone's random walk model was extended for different situations by Laming [12]. These two models accumulate the log-likelihood ratio of the evidence for each option until the accumulated samples exceed a threshold (see [13] for a detailed tutorial on the sequential sampling ratio test and how it is connected to a random walk model). The main advantage of models following the sequential sampling ratio test is their ability to specify a level of accuracy in a minimum time, making it optimal concerning reward rate when taking decision time into account [14]. In 1975, Link and Heath proposed the relative judgment theory," in which the accumulator accumulates the magnitudes of the noisy evidence [15]. In this theory, the accumulation process is considered a discrete-time process.

Three years later, in 1978, Ratcliff [2] suggested using a continuous-time process for the information accumulation and proposed a diffusion model for binary decision tasks named "diffusion decision model" (DDM). This model assumes that the information is sampled from the stimulus with Brownian motion like Eq. (1.1) with a constant drift rate that fluctuates between two absorbing boundaries. Comparisons between the mentioned proposed models and some other computational models like the Poisson counter model illustrate that the diffusion model has some advantages in predicting response times and choice behavior compared to alternative models [4,16].

Since then, various extensions of the DDM have been proposed by different researchers. For example, in [17], the authors have added trial-to-trial variability to the model. Adding decay to the drift rate was another critical extension of the diffusion model that made the diffusion model more compatible with the neural basis of the decision making. The "decision field theory" [18,19] assumes an Ornstein–Uhlenbeck process as the information accumulation process in which the drift rate has a decay parameter. This model has interesting theoretical properties but is limited to binary decision tasks, so it has been extended to multi-alternative choice [20]. Then, in 2001, the leaky competing accumulator model was introduced to increase the neurological compatibility of the SSMs [21]. This model provides a framework based on the accumulator framework (i.e., each option has an accumulator for itself) in which the accumulator has an inhibitory effect on the other accumulators, and also, each accumulator has a decay mechanism. These properties of the leaky competing accumulator model make it an attractive alternative model for investigating the underlying processes of human decision making. Time-varying diffusion models are the next generation of SSMs developed for predicting the response time distribution more precisely [22]. In other words, although the diffusion model is able to predict the mean response times of correct responses, there exist some situations in which this model cannot provide an accurate prediction of the response time distribution. During the last decade, various forms of time-varying diffusion models like collapsing boundaries models [23,24] and urgency signal models [22,25] have been proposed to account for empirical data more precisely.

By introducing more complex models, the question of how the models and its parameter can be estimated became more important. In other words, there was a tradeoff between the complexity of the computational model and how well its parameters could be recovered. Consequently, some simplified computational models were introduced to solve this issue. The first model was the EZ-diffusion decision model [26]. This model simplifies the diffusion decision model by removing the starting point bias parameter and the across-trial variability parameter and obtaining an exact formulation for each remaining parameter. In a similar work [27], an accumulator model was suggested to removes the inter-trial variability (i.e., the inter-trial noise) of the accumulation process ( ) and adds the across-trial variability (i.e., allowing drift rate changes from trial to trial) to the model.

Other developments of the SSMs continued by adding different mechanisms to the previously proposed models. For example, the attentional drift-diffusion model [28] added an attention mechanism to the diffusion decision model based on the gaze allocation on each option. This model assumes that the decision maker samples information from the currently fixated option. In the general drift-diffusion model [29], a diffusion model is suggested in which the boundaries or the drift rate can be time-dependent. The main advantage of the general drift-diffusion model is that the behavior of the across-trial variability distributions, drift rate, and thresholds are not specified, and we can select any function for them. Another extension of the diffusion models is the circular diffusion model, which provides a framework for handling the decisions with continuous option space [30]. In this model, the accumulation is done in a two-dimensional space surrounded by a circle, and each point on this circle corresponds to an option.

In more recent developments of the SSMs, some powerful models based on the framework of the accumulator models have been introduced. For example, the advantage linear ballistic accumulator model assumes that each accumulator corresponds to integrating the benefits of an option in comparison with another option [31], which provides a mathematically tractable framework for combining the inhibition ability with the accumulator framework. Alternatively, in another research, the inner clock is considered an extra accumulator in the time-based race diffusion model [32], and adding this extra accumulator to the model makes it suitable for decision making under time pressure. In [5] some more details on the history of SSMs are provided. Fig. 1.2 exhibits an overview of some significant contributions in the field of SSMs. Although many computational frameworks incorporate various cognitive mechanisms of decision making, there is still a need for theory advancement and model comparison to understand the cognitive processes underlying decision making