We all know that correlations range from - 1 to +1. What about correlations between random variables taking only on positive values, possibly from a Poisson, Exponential or Gamma joint distribution? You would think that that these multivariate  random variables  have a hard time having a very negative correlation. Here we focus on a specific example that has practical applications.

Let's assume that we are dealing with a bivariate distribution (X, Y), with the two marginals X and Y having an exponential distribution. What is the most negative correlation that we could have between X and Y? The answer is not -1, indeed it's about -0.645, and the exact value is 1 - (Pi^2) /6. Read this article for a proof, and for more general results. In particular, if you want to generate an even more negative correlation, try with Gamma distributions.

Application

This model has been used for weather predictions: the variables X and Y being respectively the storm cells duration and intensity, typically modeled as independent variables, while actually, the more intense the precipitations, the shorter the duration (thus a negative correlation). So this problem helps develop a more accurate weather prediction system. You can read the detailed paper here. 

Related articles:

• Mars Craters: An Interesting Stochastic Geometry Problem - Also a beautiful mathematical problem involving spatial statistics
• Our past mathematical challenges - One unresolved problem still offering a $500,000 award

DSC Resources

• Career: Training | Books | Cheat Sheet | Apprenticeship | Certification | Salary Surveys | Jobs
• Knowledge: Research | Competitions | Webinars | Our Book | Members Only | Search DSC
• Buzz: Business News | Announcements | Events | RSS Feeds
• Misc: Top Links | Code Snippets | External Resources | Best Blogs | Subscribe | For Bloggers

Additional Reading

• What statisticians think about data scientists
• Data Science Compared to 16 Analytic Disciplines
• 10 types of data scientists
• 91 job interview questions for data scientists
• 50 Questions to Test True Data Science Knowledge
• 24 Uses of Statistical Modeling
• 21 data science systems used by Amazon to operate its business
• Top 20 Big Data Experts to Follow (Includes Scoring Algorithm)
• 5 Data Science Leaders Share their Predictions for 2016 and Beyond
• 50 Articles about Hadoop and Related Topics
• 10 Modern Statistical Concepts Discovered by Data Scientists
• Top data science keywords on DSC
• 4 easy steps to becoming a data scientist
• 22 tips for better data science
• How to detect spurious correlations, and how to find the real ones
• 17 short tutorials all data scientists should read (and practice)
• High versus low-level data science

Follow us on Twitter: @DataScienceCtrl | @AnalyticBridge