Very nice! I actually used the featured example from Mark Richardson's class notes on Principal Component Analysis (http://people.maths.ox.ac.uk/richardsonm/SignalProcPCA.pdf) in teaching. It was astounding how clear it was to some people and how unclear to others.
I did a singular value decomposition on a data set similar to the one Richardson used (except with international data). The original post here looks at the projection to country-coordinates, looking at what axes describe primary differences between countries. My students had no problem with that -- Wales and North Ireland are most different, in your example, and 'give' the first principal axis. But then I continued to do it with the foods, as Richardson did (look at Figure 4 in the linked file). Students concluded in large numbers that people just don't like fresh fruit and do like fresh potatoes. Hm. They didn't conclude that people don't like Wales and do like North Ireland; they accurately saw it as an axis. But once we were talking about food instead of countries, students saw projection to the eigenspace as being indicative of some percentage of approval.
How could we visually display both parts of this principal component analysis to combat this prejudice that sometimes leads us to read left to right as worse to better?
By labelling the axes longitude and latitude ;) Or you could show the mean first and explain that the mean describes which foodstuffs are popular, and PC1 / PC2 refers to deviations from that mean.
Comments
Very nice! I actually used the featured example from Mark Richardson's class notes on Principal Component Analysis (http://people.maths.ox.ac.uk/richardsonm/SignalProcPCA.pdf) in teaching. It was astounding how clear it was to some people and how unclear to others.
I did a singular value decomposition on a data set similar to the one Richardson used (except with international data). The original post here looks at the projection to country-coordinates, looking at what axes describe primary differences between countries. My students had no problem with that -- Wales and North Ireland are most different, in your example, and 'give' the first principal axis. But then I continued to do it with the foods, as Richardson did (look at Figure 4 in the linked file). Students concluded in large numbers that people just don't like fresh fruit and do like fresh potatoes. Hm. They didn't conclude that people don't like Wales and do like North Ireland; they accurately saw it as an axis. But once we were talking about food instead of countries, students saw projection to the eigenspace as being indicative of some percentage of approval.
How could we visually display both parts of this principal component analysis to combat this prejudice that sometimes leads us to read left to right as worse to better?
By labelling the axes longitude and latitude ;) Or you could show the mean first and explain that the mean describes which foodstuffs are popular, and PC1 / PC2 refers to deviations from that mean.