In the first five chapters of this book, we introduced two main families of low-dimensional models for high-dimensional data: sparse models and low-rank models. In Chapter 5, we saw how we could combine these basic models to accommodate data matrices that are superpositions of sparse and low-rank matrices. This generalization allowed us to model richer classes of data, including data containing erroneous observations. In this chapter, we further generalize these basic models to a situation in which the object of interest consists of a superposition of a few elements from some set of “atoms” (Section 6.1). This construction is general enough to include all of the models discussed so far, as well as several other models of practical importance. With this general idea in mind, we then discuss unified approaches to studying the power of low-dimensional signal models for estimation, measured in terms of the number of measurements needed for exact recovery or recovery with sparse errors (Section 6.2). These analyses generalize and unify the ideas developed over the earlier chapters, and offer definitive results on the power of convex relaxation. Finally, in Section 6.3, we discuss limitations of convex relaxation, which in some situations will force us to consider nonconvex alternatives, to be studied in later chapters.

This topic examines not just what goods and services consumers buy, but why they buy them. The standard neoclassical model, based on expected utility theory and indifference curve analysis, examines the ‘what’ question, but is too narrow in focus and involves numerous anomalies. To gain a better understanding of what people buy it is necessary to understand the psychology of consumers and examine the ‘why’ question. This approach reveals several important biases which cause the anomalies: biases in expectations, biases in estimating and maximizing utilities and biases in discounting. These biases are often a result of bounded rationality, social preferences and emotional or visceral influences. The field of behavioural economics has developed a body of theory, based on the concepts of prospect theory and mental accounting, which accounts for these biases and anomalies. The fundamental concepts here are the use of reference points and loss aversion. These and other behavioural factors are in turn based on evolutionary psychology. The process of natural selection has caused our brains to evolve not as utility maximising machines but as biological fitness maximising machines.

Chapter 6 describes the first-order reliability method (FORM), which employs full distributional information. The chapter begins with a presentation of the important properties of the outcome space of standard normal random variables, which are used in FORM and other reliability methods. The FORM is presented as an approximate method that employs linearization of the limit-state surface at the design point in the standard normal space. The solution requires transformation of the random variables to the standard normal space and solution of a constrained optimization problem to find the design point. The accuracy of the FORM approximation is discussed, and several measures of error are introduced. Measures of importance of the random variables in contributing to the variance of the linearized limit-state function and with respect to statistically equivalent variations in means and standard deviations are derived. Also derived are the sensitivities of the reliability index and the first-order failure probability approximation with respect to parameters in the limit-state function or in the probability distribution model. Other topics in this chapter include addressing problems with multiple design points, solution of an inverse reliability problem, and numerical approximation of the distribution of a function of random variables by FORM.

This topic examines relationships between inputs and outputs in the production process. The starting point is the explanation of various concepts that are fundamental in production and costs: factors of production, fixed and variable factors, short and long run, scale, productivity and efficiency. The increasing importance of intangible factors is discussed, along with their main features of scalability, sunkenness, spillover effects and synergy. Production functions can take various mathematical forms. The significance of average and marginal product is explained. The concepts of the law of diminishing returns, isoquants, economies of scale, diseconomies of scale and returns to scale are introduced and interpreted. The determination of the optimal use of factors of production is explained, using mathematical analysis. Case studies play an important role in this topic in terms of demonstrating the application of theoretical concepts to real-life situations, particularly in a digital age where intangible assets and network effects are important. The leading case study relates to the so-called ‘productivity puzzle’, and various explanations of the puzzle are presented.

The problem of identifying low-dimensional structure of signals or data in high-dimensional spaces is one of the most fundamental problems that, through a long history, interweaves many engineering and mathematical fields such as system theory, pattern recognition, signal processing, machine learning, and statistics.

Chapter 10 describes Bayesian methods for parameter estimation and updating of structural reliability in the light of observations. The chapter begins with a description of the sources and types of uncertainties. Uncertainties are categorized as aleatory or epistemic; however, it is argued that this distinction is not fundamental and makes sense only within the universe of models used for a given project. The Bayesian updating formula is then developed as the product of a prior distribution and the likelihood function, yielding the posterior (updated) distribution of the unknown parameters. Selection of the prior and formulation of the likelihood are discussed in detail. Formulations are presented for parameters in probability distribution models, as well as in mathematical models of physical phenomena. Three formulations are presented for reliability analysis under parameter uncertainties: point estimate, predictive estimate, and confidence interval of the failure probability. The discussion then focuses on the updating of structural reliability in the light of observed events that are characterized by either inequality or equality expressions of one or more limit-state functions. Also presented is the updating of the distribution of random variables in the limit-state function(s) in the light of observed events, e.g., the failure or non-failure of a system.

This second chapter on business strategy examines marketing mix decisions. In particular, it focuses on more complex aspects of pricing not covered in the context of Chapter 9 on market structure and pricing. This starts with a discussion of price discrimination, and its various degrees and types. It then moves on to examine multi-product pricing, transfer pricing and dynamic aspects of pricing over the product life cycle (PLC). A detailed discussion of psychological pricing is included, and this covers behavioural aspects not normally considered within the scope of managerial economics, but highly prominent in real-life applications. Advertising decisions are also discussed in the context of the marketing mix, examining the different strategy variables in terms of content and choice of media. Recent trends in strategy related to digital and social media are discussed. There is also a discussion on the controversial topic of the effects of advertising on welfare. Finally, there is an advanced section at the end of the chapter related to optimising the marketing mix, which is mathematical in nature and involves some counterintuitive conclusions.

The Laplace transform is a mathematical operation that converts a function from one domain to another. And why would you want to do that? As you’ll see in this chapter, changing domains can be immensely helpful in extracting information from the mathematical functions and equations that describe the behavior of natural phenomena as well as mechanical and electrical systems. Specifically, when the Laplace transform operates on a function f(t) that depends on the parameter t, the result of the operation is a function F(s) that depends on the parameter s. You’ll learn the meaning of those parameters as well as the details of the mathematical operation that is defined as the Laplace transform in this chapter, and you’ll see why the Fourier transform can be considered to be a special case of the Laplace transform.

In this chapter, we consider a form of low-dimensional structure that arises in many applications in scientific data analysis: we consider datasets consisting of a few basic motifs, repeated at different locations in space and/or time.

This topic examines the various concepts related to the costs of a firm, and in particular relationships between costs and output, and why these relationships are important for managerial decision making. The starting point is a discussion of the types of cost that are relevant, and irrelevant, for decision making. Distinctions are drawn between explicit and implicit costs, historical and current costs, sunk and incremental costs, and private and social costs. Cost relationships with output, and various types of unit cost, are explained, and the reasons for these relationships based on production theory. Distinctions between short-run cost relationships and long-run cost relationships are explained, and factors that can cause costs to change, aided by graphical presentation. Economies and diseconomies of scale, and returns to scale, are explained in cost terms, along with economies of scope. Cost-volume-profit (CVP) analysis is discussed, and its implications for managerial decision making. There is an extensive problem-solving section with many different types of example. More complex CVP problems are presented in case studies, for example battery charging for electric vehicles.