Multidimensional scaling (MDS)
Department of Mathematics and Statistics Professor Mayer Alvo makes use of heat maps in a novel way to visualize complex data arising from rankings.
Multidimensional scaling (MDS) is a mathematical technique used to enable one to visualize the similarities of individual high dimensional data points in a dataset. The algorithm provides a low dimensional graphical display of the points such that the between subject distances are preserved as much as possible. In a two-dimensional display for example, a matrix is constructed. In our context, the data represents the rankings of 55 movies by each of 5,625 rankers. When the number of data points is very large as in this case, MDS does not provide a very informative display.
Heat maps provide a novel graphical display of data appearing in a matrix. The individual vectors of the matrix are color-coded, thereby making the display more appealing. When combined with MDS, heat maps can produce very informative graphs.
More details are available in Professor Alvo’s recently published book co-authored with Professor Philip Yu and entitled Statistical Methods for Ranking Data (Springer)
About the image: In a movie rating data set, there were 72,979 possibly incomplete and tied rankings of 55 movies made by 5,625 raters. Applying a two-dimensional multidimensional scaling (MDS) technique using a new normalized Kendall distance applied to the rankings, we obtained a scatterplot of 5,625 points for the movie raters shown above on the left. However, the points are too densely clustered thereby making the scatterplot ineffective for visualizing the patterns of the ranking data. Kernel smoothing was then used to produce a heat map shown above on the right. The right half of the cloud appears to show raters who appreciate action films such as the Star Wars series and The Matrix. On the left half, the upper part consists mainly of raters interested in romance such as Casablanca and The Graduate. The lower half exhibits raters who enjoy drama movies such as Seven Samurai and To Kill a Mockingbird.