Seven (plus or minus two) Clusters, A Monte Carlo Study Larry Hoyle, Policy Research Institute, The University of Kansas

1972 Kansas Statistical Abstract

Shading by Overprinting

Shading by Line Spacing

Line Shading Detail

What did they have in common? Neither method is “continuous” So both methods required grouping or classes Fixed number of combinations Characters on a fixed grid Integer number of lines in the polygon Lines are relatively coarse

How to Group for Shading Equal Intervals Equal numbers (quantiles) By clusters Don’t group (unclassed)

Population Density – 7 Equal Intervals 100 counties fall into the bottom class

Population Density - Equal Numbers 15 counties in each class - a very different picture

Population Density - Cluster Means Group around the 7 values that “best” represent the data

Population Density - Unclassed No classes, just shade in proportion to value

Clustering Tries for “Best” grouping Each member of cluster can be represented by the mean of the group

Proc Fastclus You specify the number of clusters Minimizes cluster sum of squared distance (e.g. minimum within cluster variance) inspired by: – k-means (MacQueen) leader algorithm (Hartigan)

Example clustering - data

4 clusters y cluster data. x R-squared=.9912

4 clusters data Correlation.9956 R-squared=.9912

3 clusters y cluster data. x R-squared=.9609

How many clusters is enough?

Plot R-squared by number of clusters Sample of 300 observations, Uniform distribution, 11 cluster analyses

What happens if there really aren’t any clusters? Let’s try 500 samples

Uniform, 300 obs. per sample 500 samples, 11 clusterings each

Uniform, 1000 obs. per sample 500 samples, 11 clusterings each

Normal, 300 obs. per sample 500 samples, 11 clusterings each

Normal, 1000 obs. per sample 500 samples, 11 clusterings each

Exponential, 300 obs. per sample 500 samples, 11 clusterings each

Distribution of worst sample

Exponential, 1000 obs. per sample 500 samples, 11 clusterings each

So What’s with 7  2?

Uniform, 7  samples, 11 clusterings each

Normal, 7  samples, 11 clusterings each

Exponential, 7  samples, 11 clusterings each

Minimum R squared by sample size and distribution At least 95% of the variance for all

Histograms Equal intervals Number of observations in each interval

Needle Plot of Cluster Means

Bar chart needs more bars

The Magical Number Seven, Plus or Minus Two: Some Limits on our Capacity for Information Processing George Miller, The Psychological Review 1956, vol.63 pp

Limits on Categories for Absolute Judgments Pitch 6 Loudness 5 Visual position 9 Size of a square 5 Hue 8 Name the colors in this slide

“And finally, what about the magical number seven?” George A. Miller

Miller – Quote 1 seven wonders of the world seven seas seven deadly sins seven daughters of Atlas in the Pleiades seven ages of man seven levels of hell seven primary colors seven notes of the musical scale seven days of the week” “What about the

Miller – Quote 2 seven-point rating scale seven categories for absolute judgment seven objects in the span of attention seven digits in the span of immediate memory” “What about the

“…Perhaps there is something deep and profound behind all these sevens, something just calling out for us to discover it.” Miller – Quote 3

Miller - close “But I suspect that it is only a pernicious, Pythagorean coincidence.”

Coincidence or Nature’s Parsimony? Does our capacity match what’s needed for 95% of the variance? 95%? Hmmmm……. confidence intervals an A 19 fingers and toes 970,000 web pages Larry Hoyle Policy Research Institute University of Kansas