## Visualizing sensitivity

##### May 06, 2007

I'm a loyal reader who hopes you'll indulge him in just one or two questions.

In finance (valuation, specifically), we often create two-way sensitivity tables. Unfortunately, a three-way sensitivity table is what's most often called for. Of course, we work around this by producing multiple two-way tables.

Now, obviously, it's pretty hard to build  three-way table or chart in two dimensions, and the use-bigger-bubbles method doesn't really make sense in this kind of application-- but can you conceive of a good way to present the data in any other form?

Like he indicated, we typically see multiple two-way data tables for such data.  The virtue of this approach is that the data is exceptionally well-organized; it's great for looking up the outcome given the three dimensions (I called them Red, Green and Blue to protect the innocent.)

Further, starting from a baseline i.e. a particular cell in the table, it's easy to move our eyes up, down or jump tables to observe the impact of changing dimensions (so-called sensitivity analysis).

These data tables facilitates "local" sensitivity analysis but obscure "global" sensitivity: staring at those numbers, we feel lost in the trees and can't see the forest.  What's the effect of increasing Green on average?  What's the effect of increasing Green while decreasing Blue? etc. etc.

The junkart construct (right) is made to address these questions.  The black stripes establish the baseline, the overall range of values.  Then, if interested in the effect of Red = 0.11, we can compare those red stripes with the black.  Since the spread is wide, we note that Red = 0.11 is not a strong indicator of value, and to the extent it is, it points to lesser values.

What about Red = 0.11 and Green = 2?  Now, we focus on the first red stripes and the first green stripes.  We note that the overlapping region (which is where both conditions apply) is highly concentrated to the low end of value range.  Thus, we conclude that under those conditions, value is low (below 10,000) and further, that it is low primarily because Green = 2.

On and on for any one-way, two-way or three-way effects.

Although it's not the purpose of the chart, local sensitivity can also be observed.  For example, the highest value comes from Red = 0.09, Green = 16 and Blue = 0.30.  What if Blue decreases to 0.28?  We start on the Blue = 0.28 layer; going from right to left, as we see a blue stripe, we scan vertically to find the corresponding red and green stripes; the 3rd stripe from the right, we find the scenario of interest.  Such analysis would benefit from adding an interactive vertical guiding line.

Do you prefer 3-D plots?  Contour plots? Feel free to share your ideas!

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I tried out Excel's new heatmaps and they look pretty good: http://www.westnet.net.au/balson/Temp/cht1.PNG

and this approach isn't bad either:
http://www.westnet.net.au/balson/Temp/cht2.PNG

Dermot: Thanks for the examples.

The "heatmap" and contour plots certainly improve upon the data tables but you probably noticed that the value dimension got lost in the mix. The colors can't show the scale of the value dimension. Depending on the colors chosen, the order of the color and the width of each color "bin", many different versions of these plots can be created.

What is clear is that the small multiples principle forces us to pick one dimension to be spread across graphs (here, the Red dimension), and in some cases as in here, there is no obvious candidate for it.

When I worked as a research scientist, and later as production engineer, I would do the small multiple thing (I didn't know it had a name at the time) making several different views, so I could vary the dimension which was spread across the charts. I didn't have to pick. For each experiment or set of process conditions, a different arrangement might best show the behavior. My predecessor in one job would stare at a table of numbers and come up with his guidelines, whereas I drew a couple pages of charts and refuted all of his work, and I could use the charts to explain it to others in the group.

At first i was really confused. But understanding every section clears everything out.

How did you make your so called "junkart" visualization?

Matt: That was done in R where one can control every little aspect of a chart.

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