This last chapter summarizes most of the material in this book in a range of concluding statements. It provides a summary of the lessons learned. These lessons can be viewed as guidelines for research practice.

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.

This chapter introduces the continuous-time Fourier transform (CTFT) and its properties. Many examples are presented to illustrate the properties. The inverse CTFT is derived. As one example of its application, the impulse response of the ideal lowpass filter is obtained. The derivative properties of the CTFT are used to derive many Fourier transform pairs. One result is that the normalized Gaussian signal is its own Fourier transform, and constitutes an eigenfunction of the Fourier transform operator. Many such eigenfunctions are presented. The relation between the smoothness of a signal in the time domain and its decay rate in the frequency domain is studied. Smooth signals have rapidly decaying Fourier transforms. Spline signals are introduced, which have provable smoothness properties in the time domain. For causal signals it is proved that the real and imaginary parts of the CTFT are related to each other. This is called the Hilbert transform, Poisson’’s transform, or the Kramers–Kronig transform. It is also shown that Mother Nature “computes” a Fourier transform when a plane wave is propagating across an aperture and impinging on a distant screen – a well-known result in optics, crystallography, and quantum physics.

Common time series models allow for a correlation between observations that is likely to be largest for points that are close together in time. Adjustments can be made, also, for seasonal effects. Variation in a single spatial dimension may have characteristics akin to those of time series, and comparable models find application there. Autoregressive models, which make good intuitive sense and are simple to describe, are the starting point for discussion; then moving on to autoregressive moving average with possible differencing. The "forecast" package for R has mechanisms that allow automatic selection of model parameters. Exponential smoothing state space (exponential time series or ETS) models are an important alternative that have often proved effective in forecasting applications. ARCH and GARCH heteroskedasticity models are further classes that have been developed to handle the special characteristics of financial time series.

The brain must make inferences about, and decisions concerning, a highly complex and unpredictable world, based on sparse evidence. An “ideal” normative approach to such challenges is often modeled in terms of Bayesian probabilistic inference. But for real-world problems of perception, motor control, categorization, language understanding, or commonsense reasoning, exact probabilistic calculations are computationally intractable. Instead, we suggest that the brain solves these hard probability problems approximately, by considering one, or a few, samples from the relevant distributions. Here we provide a gentle introduction to the various sampling algorithms that have been considered as the approximation used by the brain. We broadly summarize these algorithms according to their level of knowledge and their assumptions regarding the target distribution, noting their strengths and weaknesses, their previous applications to behavioural phenomena, as well as their psychological plausibility.

Quantifying the multiscale hydraulic heterogeneity in aquifers and their effects on solute transport is the task of this chapter. Using spatial statistics, we explain how to quantify spatial variability of hydraulic properties or parameters in the aquifer using the stochastic or random field concept. In particular, we discuss spatial covariance, variogram, statistical homogeneity, heterogeneity, isotropy, and anisotropy concepts. Field examples complement the discussion. We then present a highly parameterized heterogeneous media (HPHM) approach for simulating flow and solute transport in aquifers with spatially varying hydraulic properties to meet our interest and observation scale. However, our limited ability to collect the needed information for this approach promotes alternatives such as Monte Carlo simulation, zonation, and equivalent homogeneous media (EHM) approaches with macrodispersion approaches. This chapter details the EHM with the macordispersion concept.

This chapter first reviews the linear first-order non-homogeneous ordinary differential equation. Introduction to statistics and stochastic processes follows. Afterward, it explains the stochastic fluid continuum concept, associated control volume, and spatial- and ensemble-representative control volume concepts. It then uses the well-known solute concentration definition as an example to elucidate the volume- and spatial-, ensemble-average, and ergodicity concepts. This chapter is to provide basic mathematics and statistics knowledge necessary to comprehend the themes of this book. Besides, this chapter’s home works demonstrate the power of the widely available Microsoft Excel for scientific investigations.

Delving into the specifics of spatial and temporal analytics, this chapter explores topics such as spatial neighborhood and temporal evolution of large amounts of network traffic data.

This chapter discusses some basic concepts quantifying the characteristics of functions with random fluctuations. Deterministic functions are discussed first in order to introduce functions with randomness and ways to quantify them. The concepts of phase space, ensemble mean, ergodic process, moments, covariance functions, and correlation functions are discussed briefly.

This chapter gives a short summary of mathematical instruments required to model sensor systems in the presence of both deterministic and random processes. The concepts are organized in a compact overview for a more rapid consultation, emphasizing the convergences between different contexts.

Cross-cultural methodology has made significant progress. Yet, research methodology is in part a response to the challenges posed by research agendas given the intellectual and technological resources available at the time. Just as the contemporary achievements in cross-cultural research methodology are a response to the necessity of the synchronic cross-cultural research in the globalizing world of the late 20th century, the changing cultural landscape of the early 21st century demands a diachronic research agenda – cross-temporal research of psychological phenomena in human populations. Meaning equivalence and autocorrelation are highlighted as major challenges to both synchronic and diachronic approaches in cross-cultural psychology. Researchers need to meet these challenges in the future.