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1.1 Computational Statistics and Data Analysis

To do data analysis is to do computing. Statisticians have always been heavy users of whatever computing facilities are available to them. As the computing facilities have become more powerful over the years, those facilities have obviously decreased the amount of effort the statistician must expend to do routine analyses. As the computing facilities have become more powerful, an opposite result has occurred, however; the computational aspect of the statistician's work has increased. This is because of paradigm shifts in statistical analysis that are enabled by the computer.

Statistical analysis involves use of observational data together with domain knowledge to develop a model to study and understand a data-generating process. The data analysis is used to refine the model or possibly to select a different model, to determine appropriate values for terms in the model, and to use the model to make inferences concerning the process. This has been the paradigm followed by statisticians for centuries. The advances in statistical theory over the past two centuries have not changed the paradigm, but they have improved the specific methods. The advances in computational power have enabled newer and more complicated statistical methods. Not only has the exponentially-increasing computational power allowed use of more detailed and better models, however, it has shifted the paradigm slightly. Many alternative views of the data can be examined. Many different models can be explored. Massive amounts of simulated data can be used to study the model/data possibilities.

When exact models are mathematically intractable, approximate methods, which are often based on asymptotics, or methods based on estimated quantities must be employed. Advances in computational power and developments in theory have made computational inference a viable and useful alternative to the standard methods of asymptotic inference in traditional statistics. Computational inference is based on simulation of statistical models.

The ability to perform large numbers of computations almost instantaneously and to display graphical representations of results immediately has opened many new possibilities for statistical analysis. The hardware and software to perform these operations are readily available and are accessible to statisticians with no special expertise in computer science. This has resulted in a two-way feedback between statistical theory and statistical computing. The advances in statistical computing suggest new methods and development of supporting theory; conversely, the advances in theory and methods necessitate new computational methods.

Computing facilitates the development of statistical theory in two ways. One way is the use of symbolic computational packages to help in mathematical derivations (particularly in reducing the occurrences of errors in going from one line to the next!). The other way is in the quick exploration of promising (or unpromising!) methods by simulations. In a more formal sense also, simulations allow evaluation and comparison of statistical methods under various alternatives. This is a widely-used research method. For example, out of 61 articles published in the Theory and Methods section of the Journal of the American Statistical Association in 2002,  reported on Monte Carlo studies of the performance of statistical methods. A general outline of many research articles in statistics is

1. state the problem and summarize previous work on it,

2. describe a new approach,

3. work out some asymptotic properties of the new approach,

4. conduct a Monte Carlo study showing the new approach in a favorable light.
Much of the effort in mathematical statistics has been directed toward the easy problems of exploration of asymptotic properties. The harder problems for finite samples require different methods. Carefully conducted and reported Monte Carlo studies often provide more useful information on the relative merits of statistical methods in finite samples from a range of model scenarios.

While to do data analysis is to compute, we do not identify all data analysis, which necessarily uses the computer, as ''statistical computing'' or as ''computational statistics''. By these phrases we mean something more than just using a statistical software package to do a standard analysis. We use the term ''statistical computing'' to refer to the computational methods that enable statistical methods. Statistical computing includes numerical analysis, database methodology, computer graphics, software engineering, and the computer/human interface. We use the term ''computational statistics'' somewhat more broadly to include not only the methods of statistical computing, but also statistical methods that are computationally intensive. Thus, to some extent, ''computational statistics'' refers to a large class of modern statistical methods. Computational statistics is grounded in mathematical statistics, statistical computing, and applied statistics. While we distinguish ''computational statistics'' from ''statistical computing'', the emergence of the field of computational statistics was coincidental with that of statistical computing, and would not have been possible without the developments in statistical computing.

One of the most significant results of the developments in statistical computing during the past few decades has been the statistical software package. There are several of these, but a relatively small number that are in widespread use. While referees and editors of scholarly journals determine what statistical theory and methods are published, the developers of the major statistical software packages determine what statistical methods are used. Computer programs have become necessary for statistical analysis. The specific methods of a statistical analysis are often determined by the available software. This, of course, is not a desirable situation, but, ideally, the two-way feedback between statistical theory and statistical computing dimishes the effect over time.

The importance of computing in statistics is also indicated by the fact that there are at least ten major journals with titles that contain some variants of both ''computing'' and ''statistics''. The journals in the mainstream of statistics without ''computing'' in their titles also have a large proportion of articles in the fields of statistical computing and computational statistics. This is because, to a large extent, recent developments in statistics and in the computational sciences have gone hand in hand. There are also two well-known learned societies with a primary focus in statistical computing: the International Association for Statistical Computing (IASC), which is an affiliated society of the International Statistical Institute (ISI), and the Statistical Computing Section of the American Statistical Association (ASA). There are also a number of other associations focused on statistical computing and computational statistics, such as the Statistical Computing Section of the Royal Statistical Society (RSS), and the Japanese Society of Computational Statistics (JSCS).

Developments in computing and the changing role of computations in statistical work have had significant effects on the curricula of statistical education programs both at the graduate and undergraduate levels. Training in statistical computing is a major component in some academic programs in statistics (see Gentle, 2004[4], Lange, 2004[10], and Monahan, 2004[12]) In all academic programs, some amount of computing instruction is necessary if the student is expected to work as a statistician. The extent and the manner of integration of computing into an academic statistics program, of course, change with the developments in computing hardware and software and advances in computational statistics.

We mentioned above the two-way feedback between statistical theory and statistical computing. There is also an important two-way feedback between applications and statistical computing, just as there has always been between applications and any aspect of statistics. Although data scientists seek commonalities among methods of data analysis, different areas of application often bring slightly different problems for the data analyst to address. In recent years, an area called ''data mining'' or ''knowledge mining'' has received much attention. The techniques used in data mining are generally the methods of exploratory data analysis, of clustering, and of statistical learning, applied to very large and, perhaps, diverse datasets. Scientists and corporate managers alike have adopted data mining as a central aspect of their work. Specific areas of application also present interesting problems to the computational statistician. Financial applications, particularly risk management and derivative pricing, have fostered advances in computational statistics. Biological applications, such as bioinformatics, microarray analysis, and computational biology, are fostering increasing levels of interaction with computational statistics.

The hallmarks of computational statistics are the use of more complicated models, larger datasets with both more observations and more variables, unstructured and heterogeneous datasets, heavy use of visualization, and often extensive simulations.

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