The intricate dynamics of dependence and correlation are explored in this chapter. The discussion spans from linear to nonlinear correlations, encompassing regression and information-theoretic measures. This chapter lays the groundwork for deciphering the complex interplay between variables.

Delving into the fundamentals of machine learning, this chapter dissects supervised, unsupervised, and reinforcement learning approaches. The chapter discusses training, validation, and testing models, furnishing readers with the requisite skills for harnessing machine learning in data-driven modeling.

Expanding on estimation techniques, this chapter focuses on parametric approaches such as maximum likelihood estimation and Bayesian parameter estimation. It unveils methods to fit parametric distributions to empirical data, providing complementary modeling tools to the nonparametric ones discussed in the previous chapter.

Delving into multivariate probabilities, this chapter explores both nonparametric and parametric estimation techniques. From Pearson’s covariance matrix to multivariate normal distribution, readers gain insights into modeling relationships among multiple variables.

Starting from the cornerstone of probability theory, this chapter elucidates essential concepts such as quantiles, expected values, and moments of distribution. It lays the groundwork for understanding both univariate and multivariate probabilities, which are fundamental for probabilistic modeling endeavors.

This chapter explores the pivotal role of modeling as a conduit between diverse data representations and applications in real, complex systems. The emphasis is on portraying modeling in terms of multivariate probabilities, laying the foundation for the probabilistic data-driven modeling framework.

In this chapter, we selectively present global methods for efficiently solving FPDEs, employing the basis functions introduced in Chapters 2 and 3. Here, we adopt the term global often in the context of space-time, considering time as another (space-like) spectral direction. We examine a number of typical FPDEs, which we introduced and probabilistically interpreted in Chapter 1, including: the subdiffusion equation, tempered fractional diffusion on the half/whole line, in addition to the generalized and unified (1+d)-dimensional sub-to-superdiffusion FPDE model for d≥1, where a single FPDE form can model a range of physical processes by just varying the corresponding temporal/spatial fractional derivatives in the model, hence, rendering the FPDE elliptic, parabolic, and/or hyperbolic on the (1+d)-dimensional space-time hypercube. In this chapter, we employ one-sided, two-sided, constant/variable-order, and fully distributed order fractional operators, introduced in Chapters 1 and 2.

As highlighted in Chapter 1, anomalous transport phenomena can be observed in a wide variety of complex, multi-scale, and multi-physics systems such as: sub-/super-diffusion in subsurface transport, kinetic plasma turbulence, aging polymers, glassy materials, in addition to amorphous semiconductors, biological cells, heterogeneous tissues, and fractal disordered media. In this chapter, we focus on some selective applications of FPDEs and the methods presented in earlier chapters, reporting the scientific evidence of how and why fractional modeling naturally emerges in each case, along with a review of selected nonlocal mathematical models that have been proposed. The applications of interest are: (i) concentration transport in surface/subsurface dynamics, (ii) complex rheology and material damage, and (iii) fluid turbulence and geostrophic transport.

We initially introduce the standard diffusion model solving the PDF of the Brownian motion/process, satisfying the normal scaling property. This happens through a new definition of the process increments, where they are no longer drawn from a normal distribution, leading to α-stable Lévy flights at the microscopic level and correspondingly an anomalous diffusion model with a fractional Laplacian at the macroscopic scale. Next, we show how the Riemann–Liouville fractional derivatives emerge in another anomalous diffusion model corresponding to the asymmetric α-stable Lévy flights at small scales. Subsequently, we introduce the notion of subdiffusion stochastic processes, in which the Caputo time-fractional derivative appears in the anomalous subdiffusion fractional model. We combine the previous two cases, and construct continuous-time random walks, where a space-time fractional diffusion model will solve the evolution of the probability density function of the stochastic process. Next, we motivate and introduce many other types of fractional derivatives that will code more complexity and variability at micro-to-macroscopic scales, including fractional material derivatives, time-variable diffusivity for the fractional Brownian motion, tempered/variable-order/distributed-order/vector fractional calculus, etc.

This chapter provides a comprehensive presentation of global numerical methods for solving FODEs employing the polynomial and non-polynomial bases, introduced in Chapter 2. The FODEs of interest will be initial-/boundary-value problems, posed using a variety of fractional derivatives (e.g., Caputo, Riemann–Liouville, Riesz, one-sided, two-sided, variable-order, distributed order, etc.), introduced in Chapters 1 and 2. We devote Sections 3.1 and 3.2 to introducing a series of variational and non-variational spectral methods in single domains, where the solution singularities can occur at the initial or boundary points. In a variational formulation of an FODE, one first obtains the weak (variational) form of the given equation, where the highest derivative order is reduced using integration-by-parts, and then solves the variational formulation by constructing the corresponding (finite-dimensional) solution and test subspaces. In non-variational problems, one rather directly solves the strong (original) FODE, hence assuming a higher regularity in the solution. Moreover, we introduce spectral element methods (SEM) for FODEs in multiple domains for the main purpose of capturing possible interior/boundary singularities.

We present the need for new fractional spectral theories, explicitly yielding rather non-polynomial, yet orthogonal, eigensolutions to effectively represent the singularities in solutions to FODEs/FPDEs. To this end, we present the regular/singular theories of fractional Sturm–Liouville eigen-problems. We call the corresponding explicit eigenfunctions of these problems Jacobi poly-fractonomials. We demonstrate their attractive properties including their analytic fractional derivatives/integrals, three-term recursions, special values, function approximability, etc. Subsequently, we introduce the notion of generalized Jacobi poly-fractonomials (GJPFs), expanding the range of admissible parameters also allowing function singularities of negative indices at both ends. Next, we present a rigorous approximation theory for GJPFs with numerical examples. We further generalize our fractional Sturm–Liouville theories to regular/singular tempered fractional Sturm–Liouville eigen-problems, where a new exponentially tempered family of fractional orthogonal basis functions emerges. We finally introduce a variant of orthogonal basis functions suitable for anomalous transport that occurs over significantly longer time-periods.