Title: Mathematica, Version 7 New Release of Software by Wolfram Research, Inc.

Content: Introduction

The development of Mathematica® by Stephen Wolfram started in 1986, and the first version of Mathematica appeared in 1988. Further versions followed in 1991 (Version 2), 1996 (Version 3), 1999 (Version 4) and 2003 (Version 5). In the first two editions of his book, [2] described Mathematica as "a system for doing mathematics by computer", which, in fact, is the main field of application of Mathematica as a computer algebra system. But over the years, the functional range of Mathematica was extended also towards non-mathematical applications, and consequently, Mathematica was more recently characterized as a "fully integrated environment for technical computing" [2, p. IX]. In May, 2007, Mathematica experienced a major change with the implementation of version 6. With this version, especially a completely new graphics system was introduced, which simplifies graphic editing, and which allows to easily create interactive and animated graphs. Together with the slide show environment, this provides new opportunities for using Mathematica as a tool for teaching, see [1]. Only a year later, in November, 2008, the current version 7 of Mathematica was released. Again, one of the main innovations of this new version concerns the creation of graphs by offering tools for image processing and analysis, but also new statistical methods have been implemented.

Doing Statistics with Mathematica

From the outset, Mathematica offered methods for doing statistics and probability theory. In the field of probability theory, it provides functions for theoretically analyzing and simulating a variety of discrete and continuous, of univariate and multivariate distributions. As the empirical counterpart, a large repertoire of procedures for descriptive statistics (both univariate and multivariate) is available, which is supplemented by basic methods from inductive statistics such as simple parametric tests and confidence intervals. To be able to apply these functions to given data sets, the command Import is offered, the functional range of which was continually extended such that it allows meanwhile, for instance, to easily import data from Excel or OpenDocument spreadsheets as well as from Access files. Furthermore, the DatabaseLink` (since version 5.1) enables to connect to SQL data bases. For details, consult [1].

Over the years, this basic equipment of statistical methods was consistently extended. Version 4.2 offered a new package for the analysis of variance, version 5.0 introduced a package for generating statistical graphs like box plots, quantile plots and others; this collection of graphical tools was further enlarged with the new version 7. With version 5.1, a package for cluster analysis was implemented, and until the current version 7, the available functions for multilinear and nonlinear regression analysis were regularly modified and expanded. Especially, the powerful command GeneralizedLinearModelFit has now been added, which allows to fit GLMs for different distributions (binomial, inverse Gaussian, . . . ), with different link functions (logit, log-log, negative binomial, . . . ) and different estimators for covariance and dispersion. A variety of statistics for evaluating the goodness-of-fit are offered (including AIC, BIC and several likelihood statistics) as well as different types of residuals and diagnostic tools.

Conclusions

Mathematica is an easily operated and very powerful program for doing mathematics and much more, in particular, many statistical methods have been implemented meanwhile. So can Mathematica be considered as being at eye level with common statistical packages like SAS, STATISTICA or R? No, at least not yet. In some fields, as outlined above, Mathematica does not have to fear the competition with such packages, and regarding the available tools for probability theory or data import, Mathematica even seems to be superior in some cases. But on the other hand, Mathematica still shows big gaps in the field of statistical data analysis. Concerning explorative statistics, neither procedures from traditional disciplines such as discriminant or factor analysis are available nor techniques from data mining such as association rules mining. Concerning inductive statistics, some obvious gaps are time series analysis, nonparametric procedures or contingency tables, not to mention advanced topics such as methods from industrial statistics. The tide turns a bit if one considers that the enormous mathematical power of Mathematica allows to implement additional or completely new statistical procedures relatively easily.

References

[1] Weiß, C.H., 2008. Mathematica kompakt: Einführung -- Funktionsumfang -- Praxisbeispiele. R. Oldenbourg Verlag, München, Wien. [2] Wolfram, S., 2003. The Mathematica Book. 5th edition, Wolfram Media.

Reprinted with permission

Contact Information: Christian H. Weiß, University of Würzburg, Institute of Mathematics, Department of Statistics, Am Hubland, D-97074 Würzburg, Germany. Phone: +49 (931) 31 84968, Fax: +49 (931) 888 4949. Email address: christian.weiss@mathematik.uni-wuerzburg.de

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