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I teach an introductory economic geography course. To help my students develop a better understanding of the kinds of countries found in the contemporary world economy and an appreciation of data reduction techniques, I want to construct an assignment that creates a typology of different kinds of countries (e.g, high-income high-value added mfg long life expectancy; high income natural resource exporter mid-high life expectancy; with Germany being an element of the first type, and Yemen an example of the second type). This would use publicly available UNDP data (which if I recall correctly contains socioeconomic data on a bit less than 200 countries; sorry no regional data are available).

Prior to this assignment would be another which asks them (using the same --- largely interval or ratio level --- data) to examine correlations between these same variables.

My hope is that they would first develop an intuition for the kinds of relationships between different variables (e.g., a positive relationship between life expectancy and [various indicators of] wealth; a positive relationship between wealth and export diversity). Then, when using the data reduction technique, the components or factors would make some intuitive sense (e.g., factor / component 1 captures the importance of wealth; factor / component 2 captures the importance of education).

Given that these are second to fourth year students, often with limited exposure to analytical thinking more generally, what single data reduction technique would you suggest as most appropriate for the second assignment? These are population data, so inferential statistics (p-vlaues, etc.) are not really necessary.

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As an exploratory method, PCA is a good first choice for an assignment like this IMO. It'd also be nice for them to get exposed to it; it sounds like many of them won't have seen principal components before.

In terms of data I'd also point you to the World Bank Indicators, which are remarkably complete: http://data.worldbank.org/indicator.

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I agree with JMS, and PCA seems like a good idea after examining the initial correlations and scatterplots between the variables for each county. This thread has some useful suggestions to introduce PCA in non-mathematical terms.

I would also suggest utilizing small multiple maps to visualize the spatial distributions of each of the variables (and there are some good examples in this question on the gis.se site). I think these work particularly well if you have a limited number of areal units to compare and you use a good color scheme (like this example on Andrew Gelman's blog).

Unfortunately the nature of any "world countries" dataset I suspect would frequently result in sparse data (i.e. alot of missing countries), making geographic visualization difficult. But such visualization techniques should be useful in other situations as well for your course.

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  • $\begingroup$ +1, nice references. Comparing maps of the variables to maps of the PCA scores could be interesting too. $\endgroup$ – JMS Jun 11 '11 at 21:34
  • $\begingroup$ The link to the PCA introduction in non-mathematical terms was useful, as it helped me get a feel for the subtle difference between PCA and factor analysis. The GIS / mapping suggestions are also quite useful, as I hadn't thought about visualizing the spatial distribution of the variables. For this population of students, it would help them grasp the underlying structures to the world economy in a way that all my blah blah blah wouldn't. $\endgroup$ – rabidotter Jun 12 '11 at 2:50
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    $\begingroup$ Nice plots often beat blah blah blah :) $\endgroup$ – JMS Jun 12 '11 at 3:54
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A quick added note: Whichever of the above techniques you use, you'll want to first check the distributions of your variables since many of them will "require" that you first transform them using a logarithm. Doing so will reveal some of the relationships much better than using the original variables would.

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    $\begingroup$ +1 Normally a reply like this should just be posted as a comment, but the advice is so important here it benefits from every possible emphasis. PCA results in particular will likely be uninformative until variables are appropriately re-expressed. $\endgroup$ – whuber Dec 6 '11 at 15:40
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You may use CUR decomposition as an alternative to PCA. For CUR decomposition, you may refer to [1] or [2]. In CUR decomposition, C stands for the selected columns, R stands for the selected rows and U is the linking matrix. Let me paraphrase the intuition behind CUR decompsosition as given in [1];

Although the truncated SVD is widely used, the vectors $u_i$ and $v_i$ themselves may lack any meaning in terms of the field from which the data are drawn. For example, the eigenvector

[(1/2)age − (1/ √2)height + (1/2)income]

being one of the significant uncorrelated “factors” or “features” from a dataset of people’s features, is not particularly informative or meaningful.

The nice thing about CUR is that basis columns are actual columns (or rows) and better to interpret as opposed to PCA (which uses trancated SVD).

The algorithm given in [1] is easy to implement and you can play with it by changing the error threshold and get different number of bases.

[1] M.W. Mahoney and P. Drineas, “CUR matrix decompositions for improved data analysis.,” Proceedings of the National Academy of Sciences of the United States of America, vol. 106, Jan. 2009, pp. 697-702.

[2] J. Sun, Y. Xie, H. Zhang, and C. Faloutsos, “Less is more: Compact matrix decomposition for large sparse graphs,” Proceedings of the Seventh SIAM International Conference on Data Mining, Citeseer, 2007, p. 366.

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Depending on your objectives, classification of registries on groups might be best achieved by some clustering method. For a relatively small number of cases hierarchical clustering is usually best suited, at least in the exploratory phase, while for a more polished solution you might look to some iterative process like K-means. According to which software you're using it's also possible to use a process, which is in SPSS but I don't know where else, called two step clustering, which is fast, though opaque, and seems to give good results.

Cluster analysis yields a classification solution that maximizes variance between groups while minimizing variance inside said groups. It will also likely yield results that are easier to interpret.

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I suggest clustering on variables and on observations (separately) to shed light on the dataset. Variable clustering (say, using Spearmean $\rho^2$ as a similarity measure as in the R Hmisc package's varclus function) will help one see which variables "run together."

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Another option would be to use Self-Organizing Maps (SOM's). Any idea of what software the students will be using? I know that R, for example, has a couple of SOM implementations. SOM's may fail your "component factors make intuitive sense" test, however. (Not necessarily true with PCA, either...)

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  • $\begingroup$ Sorry for the delay in responding. Students would be using Minitab 16, which has some of the more traditional data reduction techniques mentioned above. I'll look into self-organizing maps, but I doubt if it would be appropriate for the kinds of students I get in a second year undergraduate course. $\endgroup$ – rabidotter Apr 28 '13 at 2:25

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