This topic examines how demand relationships can be estimated from empirical data. The whole process of performing an empirical study is explained, starting from model specification, through the collection of data, statistical analysis and interpretation of results. The focus is on statistical analysis and the application of regression analysis using OLS. Different mathematical forms of the regression model are explained, along with the relevant transformations and interpretations. The concept of goodness of fit, and the coefficient of determination, are explained, along with their application in selecting the best model. The advantages of using multiple regression are discussed, and its implementation and interpretation. Analysis of variance (ANOVA) is explained, and how this relates to goodness of fit. The implications of empirical studies are also discussed, and the light they shed on economic theory. More advanced aspects, related to inferential statistics and hypothesis testing, are covered in an appendix, along with the assumptions involved in the classical linear regression model (CLRM) and consequences of the violation of these assumptions.

The value of knowing the Laplace transforms of the basic functions described in the previous chapter is greatly enhanced by certain properties of the Laplace transform. That is because these properties allow you to determine the transform of much more complicated time-domain functions by combining and modifying the transforms of simple functions such as those discussed in .

Magnetic resonance imaging (MRI) is based on the science of nuclear magnetic resonance (NMR). Magnetic resonance states that certain atomic nuclei (such as the protons in water molecules) can absorb and emit radio-frequency energy when placed in an external magnetic field. The emitted energy is proportional to important physical properties of a material such as proton density. Therefore in physics and chemistry, magnetic resonance is an important method for studying structures of chemical substances, and its discoverers were awarded the Nobel Prize in Physics in 1952.

Many problems in structural reliability require the use of a computational platform, such as a finite-element code, to evaluate the limit-state function. Chapter 12 describes the framework for such coupling between a finite-element code and FORM/SORM analysis. The chapter begins with a brief review of the finite-element formulation for inelastic problems. Because FORM requires the gradients of the limit-state function, it is necessary for the finite-element code to compute not only the response vector but also its gradient with respect to selected outcomes of the random variables. The use of finite-differences for this purpose is not practical because of accuracy issues and computational demand. The direct-differentiation method (DDM) presented in this chapter provides an accurate and efficient means for this purpose. It is shown that the DDM requires a linear solution at the convergence of each iterative step in the nonlinear finite-element analysis. Next, a method for discrete representation of random fields of material properties or loads in the context of finite-element analysis is presented. The chapter concludes with a review of alternative approaches for finite-element reliability analysis or uncertainty propagation, including the use of polynomial chaos and various response-surface methods with efficient selection of experimental design points.

In man-made environments, most objects of interest are rich in regular, repetitive, symmetric structures. Figure 15.1 shows images of some representative structured objects. An image of such an object clearly inherits such regular structures and encodes rich information about the 3D shape, pose, or identity of the object.

This book is about modeling and exploiting simple structure in signals, images, and data. In this chapter, we take our first steps in this direction. We study a class of models known as sparse models, in which the signal of interest is a superposition of a few basic signals (called “atoms”) selected from a large “dictionary.” This basic model arises in a surprisingly large number of applications. It also illustrates fundamental tradeoffs in modeling and computation that will recur throughout the book.

Nonlinear stochastic dynamics is a broad topic well beyond the scope of this book. Chapter 13 describes a particular method of solution for a certain class of nonlinear stochastic dynamic problem by use of FORM. The approach belongs to the class of solution methods known as equivalent linearization. In this case, the linearization is carried out by replacing the nonlinear system with a linear one that has a tail probability equal to the FORM approximation of the tail probability of the nonlinear system – hence the name tail-equivalent linearization method. The equivalent linear system is obtained non-parametrically in terms of its unit impulse response function. For small failure probabilities, the accuracy of the method is shown to be far superior to those of other linearization methods. Furthermore, the method is able to capture the non-Gaussian distribution of the nonlinear response. This chapter develops this method for systems subjected to Gaussian and non-Gaussian excitations and nonlinear systems with differentiable loading paths. Approximations for level crossing rates and the first-passage probability are also developed. The method is extended to nonlinear structures subjected to multiple excitations, such as bi-component base motion, and to evolutionary input processes.