In practice we do not always have clear guidance from economic theory about specifying an econometric model. At one extreme, it may be said that we should “let the data speak.” It is good to know that when they “speak” that what they say makes sense. We must be aware of a particularly important phenomenon in empirical econometrics: the spurious relationship. If you encounter a spurious relationship but do not recognize it as such, you may inadequately consider such a relationship for hypothesis testing or for the creation of forecasts. A spurious relationship appears when the model is not well specified. In this chapter, we see from a case study that people can draw strong but inappropriate conclusions if the econometric model is not well specified. We see that if you a priori hypothesize a structural break at a particular moment in time, and based on that very assumption analyze the data, then it is easy to draw inaccurate conclusions. As with influential observations, the lesson here is that one should first create an econometric model, and, given that model, investigate whether there could have been a structural break.

This chapter deals with missing data and a few approaches to managing such. There are several reasons why data can be missing. For example, people can throw away older data, which can sometimes be sensible. It may also be the case that you want to analyze a phenomenon that occurs at an hourly level but only have data at the daily level; thus, the hourly data are missing. It may also be that a survey is simply too long, so people get tired and do not answer all questions. In this chapter we review various situations where data are missing and how we can recognize them. Sometimes we know how to manage the situation of missing data. Often there is no need to panic and modifications of models and/or estimation methods can be used. We encounter a case in which data can be made missing on purpose, by selective sampling, to subsequently facilitate empirical analysis. Such analysis explicitly takes account of the missingness, and the impact of missing data can become minor.

The chapter shows how classical interpolation problems of various types (Schur, Nevanlinna–Pick, Hermite–Fejer) carry over and generalize to the time-variant and/or matrix situation. We show that they all reduce to a single generalized constrained interpolation problem, elegantly solved by time-variant scattering theory. An essential ingredient is the definition of the notion of valuation for time-variant systems, thereby generalizing the notion of valuation in the complex plane provided by the classical z-transform.

Geostatistics provides tools for spatio-temporal data analysis. The subsurface application we cover in this book is sustainable farming in Denmark. Readers will learn about geophysical techniques for infering the redox conditions of the subsurface. A second application happens at the surface of the Earth: glaciers melting in Antarctica. We introduce a significant ongoing effort in radar imaging mapping of the Thwaites glacier in Antarctica. Both cases call for building spatial models from incomplete data: spatial interpolation. We cover geostatistical methods for capturing spatial variability with variograms and illustrate why variograms are essential to spatial interpolation, kriging. We introduce conditional simulation as a method for generating many interpolated maps that reproduce realistic variation. We show how these maps represent spatial uncertainty, and thereby affect prediction, such as predicting redox conditions in Danish agricultural areas. Finally, we introduce ways of spatial interpolating using training images. We show how using exisiting training image the exposed Arctic topography can help us interpolate Antarctica.

Introduces fractional Sobolev spaces, gives their most important properties and explains their connection with interpolation theory.

Chapter 5 continues the theme of digital image manipulation and considers transformations such as rotation or scaling with required pixel interpolation to create the most accurate final result. The GPU hardware texture units are used for this and their features are discussed. The cx utilities provided with our code include wrappers that significantly simplify the creation of CUDA textures. Curiously, these hardware texture units are rarely discussed in other CUDA tutorial material for scientific applications but we find they can give a 5-fold performance boost. We show how OpenCV can be used to provide a simple GUI interface for viewing the transformed images with very little coding effort. We end the chapter with a fully working 3D image registration program using affine transformations applied to volumetric MRI data sets. The 3D affine transformations are about 1500 times faster on the GPU than on the host CPU and a full registration between two MRI images of size 256 × 256 × 256 takes about one second.

This chapter explains how one can formulate nonlinear finite-volume (NFV) methods, as advanced discretization schemes, to solve the flow equation in porous media. These schemes are of particular interest because apart from being consistent, they are monotone by design. We explain the basic ideas of the NFV methods: how to construct one-sided fluxes, interpolate using harmonic averaging points, and obtain unique discrete fluxes through grid faces with convex combinations of one-sided fluxes. We outline key functions in the accompanied nfvm module in the MATLAB Reservoir Simulation Toolbox (MRST) and show some examples of how the method is applied.

Our treatment of aerodynamic performance (i.e. the mapping from shape to lift and drag for clean wings) idealized the plane as a mass point with lift and drag forces. The variation of the aerodynamic forces on the aircraft along the flight path determines its stability and the need for control with sustained authority. Addressing this issue requires an airplane model responding to gravity, thrust, and realistic aerodynamic forces and moments. A six-degree-of-freedom Newtonian rigid body model is compiled from the mass and balance properties of the airframe. Computational fluid dynamics (CFD) is used to predict the aerodynamic forces and moments, expressed in look-up tables of coefficients, and a major part of the text explains how such tables can be populated efficiently. The stability properties describe how well the aircraft recovers from external disturbances and how it reacts to commanded changes in flight attitude. The response in steady flight to small disturbances can be represented as a superposition of a small number of natural flight modes, the quantitative properties of which provide the quantified flight-handling qualities. A number of examples are given, from redesign of the Transonic Cruiser configuration for better pitch stability to CFD investigation of vortex interference on control surfaces on an unmanned aerial vehicle.​

In this chapter, the political theology of varṇāśramadharma is reintroduced. It is demonstrated that nearly all references to varṇāśramadharma and all references to dharma as a power standing above the king were introduced during the redaction of the text in the third century BCE. The various aspects of varṇāśramadharma that are found in the extant Arthaśāstra are explored in detail. Nearly all are linked to the work of the redactor. The addition of varṇāśramadharma creates a disonnance in the extant text, and the curiously hybrid character of its resulting political theory is explored.