Exploratory data analysis

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In statistics, exploratory data analysis (EDA) is an approach of analyzing data sets to summarize their main characteristics, often using statistical graphics and other data visualization methods. A statistical model can be used or not, but primarily EDA is for seeing what the data can tell us beyond the formal modeling and thereby contrasts traditional hypothesis testing. Exploratory data analysis has been promoted by John Tukey since 1970 to encourage statisticians to explore the data, and possibly formulate hypotheses that could lead to new data collection and experiments. EDA is different from initial data analysis (IDA), [1] [2] which focuses more narrowly on checking assumptions required for model fitting and hypothesis testing, and handling missing values and making transformations of variables as needed. EDA encompasses IDA.

Contents

Overview

Tukey defined data analysis in 1961 as: "Procedures for analyzing data, techniques for interpreting the results of such procedures, ways of planning the gathering of data to make its analysis easier, more precise or more accurate, and all the machinery and results of (mathematical) statistics which apply to analyzing data." [3]

Exploratory data analysis is an analysis technique to analyze and investigate the data set and summarize the main characteristics of the dataset. Main advantage of EDA is providing the data visualization of data after conducting the analysis.

Tukey's championing of EDA encouraged the development of statistical computing packages, especially S at Bell Labs. [4] The S programming language inspired the systems S-PLUS and R. This family of statistical-computing environments featured vastly improved dynamic visualization capabilities, which allowed statisticians to identify outliers, trends and patterns in data that merited further study.

Tukey's EDA was related to two other developments in statistical theory: robust statistics and nonparametric statistics, both of which tried to reduce the sensitivity of statistical inferences to errors in formulating statistical models. Tukey promoted the use of five number summary of numerical data—the two extremes (maximum and minimum), the median, and the quartiles—because these median and quartiles, being functions of the empirical distributionare defined for all distributions, unlike the mean and standard deviation; moreover, the quartiles and median are more robust to skewed or heavy-tailed distributions than traditional summaries (the mean and standard deviation). The packages S, S-PLUS, and R included routines using resampling statistics, such as Quenouille and Tukey's jackknife and Efron 's bootstrap, which are nonparametric and robust (for many problems).

Exploratory data analysis, robust statistics, nonparametric statistics, and the development of statistical programming languages facilitated statisticians' work on scientific and engineering problems. Such problems included the fabrication of semiconductors and the understanding of communications networks, which concerned Bell Labs. These statistical developments, all championed by Tukey, were designed to complement the analytic theory of testing statistical hypotheses, particularly the Laplacian tradition's emphasis on exponential families. [5]

Development

Data science process flowchart Data visualization process v1.png
Data science process flowchart

John W. Tukey wrote the book Exploratory Data Analysis in 1977. [6] Tukey held that too much emphasis in statistics was placed on statistical hypothesis testing (confirmatory data analysis); more emphasis needed to be placed on using data to suggest hypotheses to test. In particular, he held that confusing the two types of analyses and employing them on the same set of data can lead to systematic bias owing to the issues inherent in testing hypotheses suggested by the data.

The objectives of EDA are to:

Many EDA techniques have been adopted into data mining. They are also being taught to young students as a way to introduce them to statistical thinking. [8]

Techniques and tools

There are a number of tools that are useful for EDA, but EDA is characterized more by the attitude taken than by particular techniques. [9]

Typical graphical techniques used in EDA are:

Dimensionality reduction:

Typical quantitative techniques are:

History

Many EDA ideas can be traced back to earlier authors, for example:

The Open University course Statistics in Society (MDST 242), took the above ideas and merged them with Gottfried Noether's work, which introduced statistical inference via coin-tossing and the median test.

Example

Findings from EDA are orthogonal to the primary analysis task. To illustrate, consider an example from Cook et al. where the analysis task is to find the variables which best predict the tip that a dining party will give to the waiter. [12] The variables available in the data collected for this task are: the tip amount, total bill, payer gender, smoking/non-smoking section, time of day, day of the week, and size of the party. The primary analysis task is approached by fitting a regression model where the tip rate is the response variable. The fitted model is

(tip rate) = 0.18 - 0.01 × (party size)

which says that as the size of the dining party increases by one person (leading to a higher bill), the tip rate will decrease by 1%, on average.

However, exploring the data reveals other interesting features not described by this model.

What is learned from the plots is different from what is illustrated by the regression model, even though the experiment was not designed to investigate any of these other trends. The patterns found by exploring the data suggest hypotheses about tipping that may not have been anticipated in advance, and which could lead to interesting follow-up experiments where the hypotheses are formally stated and tested by collecting new data.

Software

See also

Related Research Articles

In statistics, quartiles are a type of quantiles which divide the number of data points into four parts, or quarters, of more-or-less equal size. The data must be ordered from smallest to largest to compute quartiles; as such, quartiles are a form of order statistic. The three quartiles, resulting in four data divisions, are as follows:

<span class="mw-page-title-main">Box plot</span> Data visualization

In descriptive statistics, a box plot or boxplot is a method for graphically demonstrating the locality, spread and skewness groups of numerical data through their quartiles. In addition to the box on a box plot, there can be lines extending from the box indicating variability outside the upper and lower quartiles, thus, the plot is also called the box-and-whisker plot and the box-and-whisker diagram. Outliers that differ significantly from the rest of the dataset may be plotted as individual points beyond the whiskers on the box-plot. Box plots are non-parametric: they display variation in samples of a statistical population without making any assumptions of the underlying statistical distribution. The spacings in each subsection of the box-plot indicate the degree of dispersion (spread) and skewness of the data, which are usually described using the five-number summary. In addition, the box-plot allows one to visually estimate various L-estimators, notably the interquartile range, midhinge, range, mid-range, and trimean. Box plots can be drawn either horizontally or vertically.

The five-number summary is a set of descriptive statistics that provides information about a dataset. It consists of the five most important sample percentiles:

  1. the sample minimum (smallest observation)
  2. the lower quartile or first quartile
  3. the median
  4. the upper quartile or third quartile
  5. the sample maximum
<span class="mw-page-title-main">John Tukey</span> American mathematician

John Wilder Tukey was an American mathematician and statistician, best known for the development of the fast Fourier Transform (FFT) algorithm and box plot. The Tukey range test, the Tukey lambda distribution, the Tukey test of additivity, and the Teichmüller–Tukey lemma all bear his name. He is also credited with coining the term bit and the first published use of the word software.

Uncomfortable science, as identified by statistician John Tukey, comprises situations in which there is a need to draw an inference from a limited sample of data, where further samples influenced by the same cause system will not be available. More specifically, it involves the analysis of a finite natural phenomenon for which it is difficult to overcome the problem of using a common sample of data for both exploratory data analysis and confirmatory data analysis. This leads to the danger of systematic bias through testing hypotheses suggested by the data.

<span class="mw-page-title-main">Data analysis</span> The process of analyzing data to discover useful information and support decision-making

Data analysis is the process of inspecting, cleansing, transforming, and modeling data with the goal of discovering useful information, informing conclusions, and supporting decision-making. Data analysis has multiple facets and approaches, encompassing diverse techniques under a variety of names, and is used in different business, science, and social science domains. In today's business world, data analysis plays a role in making decisions more scientific and helping businesses operate more effectively.

<span class="mw-page-title-main">Data and information visualization</span> Visual representation of data

Data and information visualization is the practice of designing and creating easy-to-communicate and easy-to-understand graphic or visual representations of a large amount of complex quantitative and qualitative data and information with the help of static, dynamic or interactive visual items. Typically based on data and information collected from a certain domain of expertise, these visualizations are intended for a broader audience to help them visually explore and discover, quickly understand, interpret and gain important insights into otherwise difficult-to-identify structures, relationships, correlations, local and global patterns, trends, variations, constancy, clusters, outliers and unusual groupings within data. When intended for the general public to convey a concise version of known, specific information in a clear and engaging manner, it is typically called information graphics.

<span class="mw-page-title-main">Frederick Mosteller</span> American statistician

Charles Frederick Mosteller was an American mathematician, considered one of the most eminent statisticians of the 20th century. He was the founding chairman of Harvard's statistics department from 1957 to 1971, and served as the president of several professional bodies including the Psychometric Society, the American Statistical Association, the Institute of Mathematical Statistics, the American Association for the Advancement of Science, and the International Statistical Institute.

In statistics the trimean (TM), or Tukey's trimean, is a measure of a probability distribution's location defined as a weighted average of the distribution's median and its two quartiles:

In statistics, the median absolute deviation (MAD) is a robust measure of the variability of a univariate sample of quantitative data. It can also refer to the population parameter that is estimated by the MAD calculated from a sample.

The median polish is a simple and robust exploratory data analysis procedure proposed by the statistician John Tukey. The purpose of median polish is to find an additively-fit model for data in a two-way layout table of the form row effect + column effect + overall median.

GGobi is a free statistical software tool for interactive data visualization. GGobi allows extensive exploration of the data with Interactive dynamic graphics. It is also a tool for looking at multivariate data. R can be used in sync with GGobi. The GGobi software can be embedded as a library in other programs and program packages using an application programming interface (API) or as an add-on to existing languages and scripting environments, e.g., with the R command line or from a Perl or Python scripts. GGobi prides itself on its ability to link multiple graphs together.

Statistical graphics, also known as statistical graphical techniques, are graphics used in the field of statistics for data visualization.

<span class="mw-page-title-main">L-estimator</span>

In statistics, an L-estimator is an estimator which is a linear combination of order statistics of the measurements. This can be as little as a single point, as in the median, or as many as all points, as in the mean.

Data Desk is a software program for visual data analysis, visual data exploration, and statistics. It carries out Exploratory Data Analysis (EDA) and standard statistical analyses by means of dynamically linked graphic data displays that update any change simultaneously.

<span class="mw-page-title-main">Plot (graphics)</span> Graphical technique for data sets

A plot is a graphical technique for representing a data set, usually as a graph showing the relationship between two or more variables. The plot can be drawn by hand or by a computer. In the past, sometimes mechanical or electronic plotters were used. Graphs are a visual representation of the relationship between variables, which are very useful for humans who can then quickly derive an understanding which may not have come from lists of values. Given a scale or ruler, graphs can also be used to read off the value of an unknown variable plotted as a function of a known one, but this can also be done with data presented in tabular form. Graphs of functions are used in mathematics, sciences, engineering, technology, finance, and other areas.

<span class="mw-page-title-main">Peter Rousseeuw</span> Belgian statistician (born 1956)

Peter J. Rousseeuw is a statistician known for his work on robust statistics and cluster analysis. He obtained his PhD in 1981 at the Vrije Universiteit Brussel, following research carried out at the ETH in Zurich, which led to a book on influence functions. Later he was professor at the Delft University of Technology, The Netherlands, at the University of Fribourg, Switzerland, and at the University of Antwerp, Belgium. Next he was a senior researcher at Renaissance Technologies. He then returned to Belgium as professor at KU Leuven, until becoming emeritus in 2022. His former PhD students include Annick Leroy, Hendrik Lopuhaä, Geert Molenberghs, Christophe Croux, Mia Hubert, Stefan Van Aelst, Tim Verdonck and Jakob Raymaekers.

Heike Hofmann is a statistician and Professor in the Department of Statistics at Iowa State University.

Dianne Helen Cook is an Australian statistician, the editor of the Journal of Computational and Graphical Statistics, and an expert on the visualization of high-dimensional data. She is Professor of Business Analytics in the Department of Econometrics and Business Statistics at Monash University and professor emeritus of statistics at Iowa State University. The emeritus status was chosen so that she could continue to supervise graduate students at Iowa State after moving to Australia.

Robust Regression and Outlier Detection is a book on robust statistics, particularly focusing on the breakdown point of methods for robust regression. It was written by Peter Rousseeuw and Annick M. Leroy, and published in 1987 by Wiley.

References

  1. ↑ Chatfield, C. (1995). Problem Solving: A Statistician's Guide (2nd ed.). Chapman and Hall. ISBN   978-0412606304.
  2. ↑ Baillie, Mark; Le Cessie, Saskia; Schmidt, Carsten Oliver; Lusa, Lara; Huebner, Marianne; Topic Group "Initial Data Analysis" of the STRATOS Initiative (2022). "Ten simple rules for initial data analysis". PLOS Computational Biology. 18 (2): e1009819. Bibcode:2022PLSCB..18E9819B. doi: 10.1371/journal.pcbi.1009819 . PMC   8870512 . PMID   35202399.
  3. ↑ John Tukey-The Future of Data Analysis-July 1961
  4. ↑ Becker, Richard A., A Brief History of S, Murray Hill, New Jersey: AT&T Bell Laboratories, archived from the original (PS) on 2015-07-23, retrieved 2015-07-23, ... we wanted to be able to interact with our data, using Exploratory Data Analysis (Tukey, 1971) techniques.
  5. ↑ Morgenthaler, Stephan; Fernholz, Luisa T. (2000). "Conversation with John W. Tukey and Elizabeth Tukey, Luisa T. Fernholz and Stephan Morgenthaler". Statistical Science. 15 (1): 79–94. doi: 10.1214/ss/1009212675 .
  6. ↑ Tukey, John W. (1977). Exploratory Data Analysis. Pearson. ISBN   978-0201076165.
  7. ↑ Behrens-Principles and Procedures of Exploratory Data Analysis-American Psychological Association-1997
  8. ↑ Konold, C. (1999). "Statistics goes to school". Contemporary Psychology. 44 (1): 81–82. doi:10.1037/001949.
  9. ↑ Tukey, John W. (1980). "We need both exploratory and confirmatory". The American Statistician. 34 (1): 23–25. doi:10.1080/00031305.1980.10482706.
  10. ↑ Sailem, Heba Z.; Sero, Julia E.; Bakal, Chris (2015-01-08). "Visualizing cellular imaging data using PhenoPlot". Nature Communications. 6 (1): 5825. Bibcode:2015NatCo...6.5825S. doi:10.1038/ncomms6825. ISSN   2041-1723. PMC   4354266 . PMID   25569359.
  11. ↑ Elementary Manual of Statistics (3rd edn., 1920)https://archive.org/details/cu31924013702968/page/n5
  12. ↑ Cook, D. and Swayne, D.F. (with A. Buja, D. Temple Lang, H. Hofmann, H. Wickham, M. Lawrence) (2007) "Interactive and Dynamic Graphics for Data Analysis: With R and GGobi" Springer, 978-0387717616

Bibliography


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