Logging a sleight of hand

Andrew puts up an interesting chart submitted by one of his readers (link):

Bruce Knuteson who created this chart is pursuing a theory that there is some fishy going on in the stock markets over night (i.e. between the close of one day and the open of the next day). He split the price data into two interleaving parts: the blue line represents returns overnight and the green line represents returns intraday (from open of one day to the close of the same day). In this example related to Tesla's stock, the overnight "return" is an eyepopping 36850% while the intraday "return" is -46%.

This is an example of an average masking interesting details in the data. One typically looks at the entire sequence of values at once, while this analysis breaks it up into two subsequences. I'll write more about the data analysis at a later point. This post will be purely about the visualization.

***

It turns out that while the chart looks like a standard time series, it isn't. Bruce wrote out the following essential explanation:

The chart can't be interpreted without first reading this note.

The left chart (a) is the standard time-series chart we're thinking about. It plots the relative cumulative percentage change in the value of the investment over time. Imagine one buys $1 of Apple stock on day 1. It shows the cumulative return on day X, expressed as a percent relative to the initial investment amount. As mentioned above, the data series was split into two: the intraday return series (green) is dwarfed by the overnight return series (blue), and is barely visiable hugging the horizontal axis.

Almost without thinking, a graphics designer applies a log transform to the vertical axis. This has the effect of "taming" the extreme values in the blue line. This is the key design change in the middle chart (b). The other change is to switch back to absolute values. The day 1 number is now $1 so the day X number shows the cumulative value of the investment on day X if one started with $1 on day 1.

There's a reason why I emphasized the log transform over the switch to absolute values. That's because the relationship between absolute and relative values here is a linear one. If y(t) is the absolute cumulative value of $1 at time t, then the percent change r(t) = 100(y(t) -1). (Note that y(0) = 1 by definition.)  The shape of the middle chart is primarily conditioned by the log transform.

In the right chart (c), which is the design that Bruce features in all his work, the visual elements of chart (b) are retained while he replaced the vertical axis labels with those from chart (a). In other words, the lines show the cumulative absolute values while the labels show the relative cumulative percent returns.

I left this note on Gelman's blog (corrected a mislabeling of the chart indices):

I'm interested in the the sleight of hand related to the plots, also tying this back to the recent post about log scales. In plot (b) (a) [middle of the panel], he transformed the data to show the cumulative value of the investment assuming one puts $1 in the stock on day 1. He applied a log scale on the vertical axis. This is fine. Then in plot (c) (b), he retained the chart but changed the vertical axis labels so instead of absolute value of the investment, he shows percent changes relative to the initial value.

Why didn't he just plot the relative percent changes? Let y(t) be the absolute values and r(t) = the percent change = 100*(y(t) -1) is a simple linear transformation of y(t). This is where the log transform creates problems! The y(t) series is guaranteed to be positive since hitting y(t) = 0 means the entire investment is lost. However, the r(t) series can hit negative values and also cross over zero many times over time. Thus, log r(t) is inoperable. The problem is using the log transform for data that are not always positive, and the sleight of hand does not fix it!

Just pick any day in which the absolute return fell below $1, e.g. the last day of the plot in which the absolute value of the investment was down to $0.80. In the middle plot (b), the value depicted is ln(0.8) = -0.22. Note that the plot is in log scale, so what is labeled as $1 is really ln(1) = 0. If we instead try to plot the relative percent changes, then the day 1 number should be ln(0) which is undefined while the last number should be ln(-20%) which is also undefined.

This is another example of something umcomfortable about using log scales which I pointed out in this post. It's this idea that when we do log plots, we can freely substitute axis labels which are not directly proportional to the actual labels. It's plotting one thing, and labelling it something else. These labels are then disconnected from the visual encoding. It's against the goal of visualizing data.

 

The message left the visual

The following chart showed up in Princeton Alumni Weekly, in a report about China's population:

This chart was one of several that appeared in a related Scientific American article.

The story itself is not surprising. As China develops, its birth rate declines, while the death rate also falls, thus, the population ages. The same story has played out in all advanced economies.

***

From a Trifecta Checkup perspective, this chart suffers from several problems.

The text annotation on the top right suggests what message the authors intended to deliver. Pointing to the group of people aged between 30 and 59 in 2020, they remarked that this large cohort would likely cause "a crisis" when they age. There would be fewer youngsters to support them.

Unfortunately, the data and visual elements of the chart do not align with this message. Instead of looking forward in time, the chart compares the 2020 population pyramid with that from 1980, looking back 40 years. The chart shows an insight from the data, just not the right one.

A major feature of a population pyramid is the split by gender. The trouble is gender isn't part of the story here.

In terms of age groups, the chart treats each subgroup "fairly". As a result, the reader isn't shown which of the 22 subgroups to focus on. There are really 44 subgroups if we count each gender separately, and 88 subgroups if we include the year split.

***

The following redesign traces the "crisis" subgroup (those who were 30-59 in 2020) both backwards and forwards.

The gender split has been removed; here, the columns show the total population. Color is used to focus attention to one cohort as it moves through time.

Notice I switched up the sample times. I pulled the population data for 1990 and 2060 (from this website). The original design used the population data from 1980 instead of 1990. However, this choice is at odds with the message. People who were 30 in 2020 were not yet born in 1980! They started showing up in the 1990 dataset.

At the other end of the "crisis" cohort, the oldest (59 year old in 2020) would have deceased by 2100 as 59+80 = 139. Even the youngest (30 in 2020) would be 110 by 2100 so almost everyone in the pink section of the 2020 chart would have fallen off the right side of the chart by 2100.

These design decisions insert a gap between the visual and the message.

 

 

Swarmed by ants

Andrew discussed the following chart in a recent blog post:

Alert! A swarm of ants has marched onto a bubble chart.

These overlapping long text labels are dominating the chart; the length of these labels encodes the length of country names, which has nothing to do with the data.

We're waiting - hoping - for the ants to march off the page.

***
Andrew's blog post is about something else, the use of log scales. The chart above is a log-log plot. Both axes have log scales.

Andrew's correspondent doesn't like log scales. Andrew does.

One problem we encounter in practice with log scales is that people without science background can't read them. Andrew's correspondent said as much, while also misinterpreting the log-log chart. He says the log-log chart "visually creates a much stronger correlation than there actually is".

But that's not what happened. It's more appropriate to say that the log transformations allow us to see the correlation that exists. The correlation is not linear which is why the usual scatter plot does not reveal it. 

Nevertheless, I agree with the correspondent on avoiding log scales in data displays because most readers don't get it.

***

Consider the following pair of plots.

The underlying data follow the pattern Y = 0.003 * X^2.5 but for what we're talking about, the specific pattern doesn't matter so long as X and Y has a "power" relationship. 

The left plot directly shows the relationship between X and Y using regular scales. Readers see that Y is running away from X. The slope of the line increases as X increases. The speed of growth of Y exceeds that of X. This relationship is curved, which can't be described in words succinctly.

The right plot visually shows a linear relationship between X and Y but it's not really between X and Y. It's between log(X) and log(Y). Note that log(Y) = log(0.003*X^2.5) = log(0.003) + 2.5*log(X), which is a straight line with slope 2.5 and intercept log(0.003). The gap between gridlines now represents a 10-fold jump in value (of X or of Y). The linear relationship is between X and Y in log scale; in linear scale, it's a power relationship, not linear.

The practice of printing axis labels in the original scale, rather than log scale, adds to the confusion. On the right plot, the points labeled 5,000 and 50,000 do not actually lie on the line; what fall in line are the points log(5,000) and log(50,000). The reason for this confusing practice is that humans have trouble understanding data in log scale. For example, if $50,000 is the GDP per capita for some country, then log($50,000) = $4.5 which can't be interpreted.

Whether we are talking about the gaps between gridlines or about specific points on the line, what readers see on the log-log chart is only part of the story. Readers must also recognize that for the log-log chart to work, equal gaps between gridlines do not signify equal gaps in the data, while the linear relationship is between the log of the axis labels, not the labels themselves.

The X-Y plot can be interpreted visually in a direct way while the log-log plot requires the reader to transcend the visual representation, entering an abstract realm.

 

 

Aligning the visual and the message

Today's post is about work by Diane Barnhart, who is a product manager at Bloomberg, and is taking Ray Vella's infographics class at NYU. The class is given a chart from the Economist, as well as some data on GDP per capita in selected countries at the regional level. The students are asked to produce data visualization that explores the change in income inequality (as indicated by GDP per capita).

Here is Diane's work:

In this chart, the key measure is the GDP per capita of different regions in Germany relative to the national average GDP. Hamburg, for example, has a GDP per capita that was 80% above the national average in 2000 while Leipzig's GDP per capita was 30% below the national average in 2000. (This metric is a bit of a head scratcher, and forms the basis of the Economist chart.)

***

Diane made several insightful design choices.

The key insight of this graph is also one of the easiest to see. It's the narrowing of the range of possible values. In 2000, the top value is about 90% while the bottom is under -40%, making a range of 130%. In 2020, the range has narrowed to 90%, with the values falling between 60% and -30%. In other words, the gap between rich and poor regions in Germany has reduced over these two decades.

The chosen chart form makes this message come alive.

Diane divided the regions into three groups, mapped to the black, red and yellow colors of the German flag. Black are for those regions that have GDP per capita above the average; yellow for those regions with GDP per capita over 25% below the average.

Instead of applying color to individual lines that trace the GDP metric over time for each region, she divided the area between the lines into three, and painted them. This necessitates a definition of the boundary line between colored areas over time. I gathered that she classified the regions using the latest GDP data (2020) and then traced the GDP trend lines back in time. Other definitions are also possible.

The two-column data table shown on the right provides further details that aren't found in the data visualization. The table is nicely enhanced with colors. They represent an augmentation of the information in the main chart, not a repetition.

All in all, this is a delightful project, and worthy of a top grade!