Even a short glance over visual data yields a wealth of information, such as the number of represented objects. The ability to grasp that number in but a moment is called subitizing. Research on the phenomenon has shown that humans can usually identify the quantity of objects up to five, and up to eight with training.

The product of two numbers adjacent to a median, is one less than the square of the median. For example, 52=25, and 6×4=25-1=24. Algebraically, this is because (n+1)(n-1)=n2-1. Geometrically

Cognitive shortcuts like subitizing are employed in the fabulous graphics made by Oliver Steele for his children. A series of block calculations serve to illustrate basic laws of algebra. These visual explanations are kin to physical bricks that see use in elementary schools and as such readily understandable. At the same time, the two-dimensional graphic design can visualize transitions far better (to my judgement at least).

It is not entirely clear to me why such trivial, yet great explanations are used so little. All the more praise for Mr. Steele and his understanding for naÏve mathematicians.

References:

“The Number Sense: How the Mind Creates Mathematics” (Stanislas Dehaene)

Most failures, most success. (Tom Peters‘ Irreducibles)

It seems that although most of my beliefs about the visual display of facts aren’t exactly widespread, there are co-conspirators. In the april issue of Juan C. Dürsteler’s newsletter on information visualization, he extends a rather unusual criticism of the widespread time axis:

Time, nevertheless is not well represented by a line since it isn’t a spatial dimension and, moreover, it’s irreversible. Paradoxically, the widest used metaphor is that of the line.

This is obviously a claim too bold, and the rebuttal by Stephen Few is entirely justified. Still, it might be quite rewarding to check visual data displays with this question in mind:

Do linear plots of time suggest either reversibility or causal disjunction where those states can really not be assumed?

It is probably easy to find examples where this is the case. In a previous post, I commented on continuous scales of text, such as in a recent visualization of the digg social news site. In that case, the timeline may suggest that future stores somehow have an effect on older ones, which of course breaks the logic of one-directional progress of time.

So in the case of a simple line graph, the error that Dürsteler points out would look something like this (fig1):

Assuming the situation of someone thinking at the point in time marked “now”,

we can assume that the data recorded before that moment, that to the left of the line (a), has some sort of influence on his thoughts.

we can however not assume any influence of anything that happens afterwards, such as (b).

Maybe time will deliver some better case scenarios. Does any reader have suggestions for a case in which time as a one-directional axis is misinterpreted so that faux causalities arise?

Numbers show facts in a very restricted way, limiting their interpretation somewhat. Still, when venturing into the realms of visualization, even number-guided visual artifacts artifacts can become ambiguous. Circumventing culturally or neurologically determined alternate ways of interpretation seems to be impossible. As a logical conclusion, a first effort must be to understand how those two systems of visual meaning extraction work, before mastering them.

A book that serves this purpose very well is “Visual Intelligence” by Donald Hoffman. In this – for a work on visual thinking – moderately priced volume, the UCI cognitive science scholar describes the way the human mind processes raw imagery data in order to create meaning, and the errors that can arise from the process. Figure 1, for example, is taken from an advertising billboard by German-based development NGO, showing the german word for bread, “Brot”, with an image in place of the letter “o”. Interestingly, it seems that people socialized with an US-inspired bread culture can see the object as a sort of fat, round roll such as those used for burgers, with some crumbs on it. Yet the intended image, according to my judgement, is one of an almost empty clay bowl with some scattered rice grains.

This is what Hoffman describes in his chapter on the way we re-establish three dimensional objects from curves. In the fig.1 example, our brain is confronted with a shape that could be reconstructed into both: a convex burger with a shadow on the outside, of a concave bowl with the shadow within.

For anyone with a strong affinity to mathematics, i should mention Hoffman’s work “Observer Mechanics” that is freely available on his web site. Be warned though, as formulas will be thrown your way here.
References:

“Visual Intelligence: How We Create What We See” (Donald David Hoffman, Donald D. Hoffman)

Obviously, basic education today covers simple types of graphs. But where to go from there? The following selection of books doesn’t go into much detail. Instead, it seeks to cover an area as broad as possible – from Tukey’s minimal methods to interactive spatial visualizations with Slocum.

#1: “Data Analysis and Regression : A Second Course in Statistics (Addison-Wesley Series in Behavioral Science)” (Frederick Mosteller, John W. Tukey) is one of the many excellent books by John Wilder Tukey, one of the figures who shaped the way we deal with data today. His groundbreaking “Exploratory Data Analysis” – written in an unexpectedly vivid, strong-minded yet playful style – is hard to come by these days. Either of these books gives our concept of statistics the basis it never really had.

#2: “The Visual Display of Quantitative Information” (Edward R. Tufte) To arrive at Edward Tufte when coming from Tukey means spilling the entire Museum of Modern Art into a book. Tufte, in his four canonized books on visual data display, has always made aesthetics a foremost priority. This goes up to a point where data may be conveyed as good as possible – but one is left asking how much data that actually was, and whether that effort really mattered in the presented case. Still, no one digs deeper and pulls up a greater variety of “visual explanations”. His latest work, “Beautiful Evidence” (Edward R. Tufte), hasn’t found its ways into my hands yet. I mention it anyways, mainly for its introduction of “sparklines”, small graphics that fit into the flow of text.

#3: “Thematic Cartography and Geographic Visualization, Second Edition” (Terry A. Slocum, Robert B McMaster, Fritz C. Kessler, Hugh H. Howard) Far too many works by statisticians omit an area that historically belongs to another field of study: Cartography. Hardly any real world data is unrelated to spatial aspects, and many graphs would greatly profit from the amassed knowledge of cartographers. “Thematic Cartography and Visualization” is a lean and efficient introduction, suitable for anyone with even the slightest knowledge of the topic.

Microsoft suggests (fig.1) that company growth is always exponential. Must be true, because they suggest to always use “an exponential chart”. (No, they don’t mean a chart with exponential scale.) The article itself hints that there may be some exceptions – but I find it highly dangerous to write this in a “classroom-style book” on financial analysis. Next time someone comes at me with an exponential chart, I won’t believe him.

In an earlier post, I proposed to differentiate naïve from skilled statisticians, and design graphs accordingly. The tool that I recommend here is very useful to display simple matters: it produces excellent graphs for naïve statisticians. It is a small, specialized, free program for Apple’s Mac Os X, and it’s called – Plot.

Just like many other mac applications, plot benefits from simplicity. It draws two-dimensional graphs: line charts, data points or histograms, and – well, that’s it. It takes data from a variety of basic file formats or – a nice feature – pull it from sql databases. What sets plot aside from more widespread solutions, like Excel, SPSS, R, Statistica and others, is its ease of use and the quality of its output (cf. example in fig.1). Once a file is loaded, line styles can be adjusted for each variable. All that is left to do is placing the legend (maybe removing graphical overhead) – et voilá, a beautiful, minimalist graph has been created in minutes.

The great advantage is that plot strikes a balance between functionality and usability: it is not as awkward to use and its output is a lot better than Excel’s, while at the same time the user interface (fig.3) stays simple. Its reductionist approach seems to be beneficial in another way: instead of trying out more and more complex visualizations (“Oh wait, maybe a 3D scatterplot could…”), the user decides on the type of chart first – and by choosing plot it will automatically be a type naïve statisticians understand.

Verdict: If I can visualize data with plot, the resulting graphs will look elegant yet precise, without superfluous distracting elements (boxes, large symbols, color background &c.) – and because of its form, I will be able to explain it to anyone without first handing him/her a statistics manual.

References:

The Plot Homepage – http://plot.micw.eu/

As more and more textual information is brought into the digital realm, inheriting structure and little descriptive tags, visualizations are put forward to combat the “Overnewsed but Underinformed” syndrome (on that note, Neil Postman should be seen as nothing but an entry point for more serious reading on media effects).

In the past years, visualizations of textual elements have come and gone. Maybe speaking of a boom is not entirely misguided ever since processing makes creating them less tedious. Among many of them, one crucial problem has caught my eye. People don’t seem to care about the scale of their variables. Text, at least to my knowledge, is still a discrete (i.e. only integer counts) nominal (i.e. no inherent order) variable.

To put this clearly: text items in their original form cannot be ordered along an axis.

There are several solutions around for this problem. All of them are less than convincing.

• One way is to just throw data points – visual representations of one textual item – at the recipient in an unsorted way, maybe with a physical simulation engine for organic movement. If someone can explain in what way the visual metaphor of fluids in a bacterium benefits visual data display, I’ll be glad to call it useful.

• Some people think it’s possible to sort text items by another nominal variable, like author or topic. That, however, does not really sort but rather group them.

• By far the most common approach is to use the continuous metadata of time and arrange by publishing date. Although this approach conveys some meaning (recency can be seen as related to importance), it pays no attention to other characteristics of text items whatsoever. A wonderful example is the new visualization (fig.1) for digg, the largest social news site, that is about to launch these days.

Analysis

Figure 1 displays the main view of “Incoming”, an upcoming visualization of activity on digg. It features a bar graph with the horizontal axis representing an adjustable time window and bar height representing story weight (number of “diggs” – for an explanation of the concept see the FAQ). There is no absolute y-axis, which makes it remarkably similar to an oversized sparkline (small data graphic propagated by Tufte – will be covered in an upcoming visual explanation of the week).

The project has been covered by Max Kiesler, his partner Emily Chang, and photos of a presentation are available on Flickr. One closeup shot reveals that the text labels below bars are actually category names (technology, gaming etc.). Translating the visual message of this display into one sentence, it says:

“Here are relative popularity and the category of 17 items that were active from 4:32:33 to 4:34:57 compared to each other.”

This in itself is a very limited statement. If we look at the digg homepage for comparison, it offers two basic modes for story display: The most popular stories accumulated from a fixed point in time up to now (Top Today, Top This Month), and the most popular stories from the last n hours (while surpassing a certain threshold, I assume, either for the absolute number of stories or popularity). This graph seems to convey the same information as the standard digg homepage, minus the title, text and other information related to stories. How exactly should users profit from this form of data display? Emily Chang states the benefits as following:

Given the sheer volume of stories and activity on Digg, it’s becoming impossible to find new, noteworthy stories or to see what was popular at any given time unless it was on the home page. This certainly gives me a method to explore and discovery the stories I’m interested in based on live activity.

To state that it is impossible to find new or noteworthy stories is of course wrong. The following two sentences do, however, get something right: this chart is about “live activity” and “any given time”. It reduces stories to their popularity rating, providing an overview of the distribution of popularity among the site’s categories at a given moment in time.

“Incoming” thus does not concern itself with the actual data (or text), it shows behavioral patterns among people casting votes for nondescript items. I assume the visualization was never designed for the public anyways, but rather as a sort of installation for the offices of digg inc.

Verdict: Aesthetic but irrelevant and source-data-agnostic.

References

“Amusing Ourselves to Death: Public Discourse in the Age of Show Business” (Neil Postman)

Most people without statistical education nevertheless have an intrinsic instinct for visual explanations. It works on at least two levels: They can usually tell whether a graph is well made, i.e. aesthetically pleasant, and they have a rough impression of what it wants to say. In reference to a great psychologist (Fritz Haider) with an absoluy uneventful life, I like to call them “Naïve Statisticians”.

Heider’s attribution theory proposed that the real workings of human mind and personality are of little interest. In their place, the crude models that ordinary people form about their likes offer powerful explanations of human behavior.

What if we take this perspective into statistics?

Through the eyes of a naïve statistician, the most complex, the most dense types of data display suddenly look the same, be they a Delaunay, or maybe flying paths in a supercollider (fig.1). The first and most important decision in any graphing, plotting and charting must therefore be to decide on the audience. Skilled thinkers need margins of error, small multiples (probably one of the next visual explanation of the week candidates), confidence intervals, many dimensions, absolute numbers. Naïve statisticians need few dimensions, zero-ed scales, comparable measurements and legends.

The need that unites them, the least common denominator, is what made Edward Tufte so famous and his books so good: Aesthetics. Keep an eye on the blank graph for reviews of tools that help.

Recommended reading:

“The Life of a Psychologist: An Autobiography” (Fritz Heider)

“The Chicago Guide to Writing about Numbers (Chicago Guides to Writing, Editing, and Publishing)” (Jane E. Miller)

“Calculated Risks: How to Know When Numbers Deceive You” (Gerd Gigerenzer)

“The Probabilistic Revolution, Volume 2 : Ideas in the Sciences (Bradford Books)” (The MIT Press)

This post probably moves away a bit from the intended focus of the blank graph, but it does so for a reason. Visualization and intelligent data analysis are to a large part about putting structure where there was none before. The human mind reaches its limits with large data sets, where many cognitively efficient ways to come to judgements fail. That is why it is important to reduce and structure.

Interestingly, a very similar approach can be applied to the way humans interact. Andreas Jungherr, a good friend of mine, (cf. my blogroll) sees stories as the integral factor in political leadership. Stories, he proposes in an article on leadership (German only so far, sorry), cut down on ambiguity, doubt and fear by condensing what’s going on in society into archetypical figures following a plot.

In this context, the philosopher Daniel Dennett has written some great descriptions of the way a good visual/textual explanation should work. The following excerpt is taken from the EDGE homepage (a recommended reading on scientific culture) which itself is one sample chapter of the book “Third Culture : Beyond the Scientific Revolution“.

Daniel C. Dennett: If you look at the history of philosophy, you see that all the great and influential stuff has been technically full of holes but utterly memorable and vivid. They are what I call “intuition pumps” — lovely thought experiments. Like Plato’s cave, and Descartes’s evil demon, and Hobbes’ vision of the state of nature and the social contract, and even Kant’s idea of the categorical imperative. I don’t know of any philosopher who thinks any one of those is a logically sound argument for anything. But they’re wonderful imagination grabbers, jungle gyms for the imagination. They structure the way you think about a problem. These are the real legacy of the history of philosophy. A lot of philosophers have forgotten that, but I like to make intuition pumps.

[…]

I coined the term “intuition pump,” and its first use was derogatory. I applied it to John Searle’s “Chinese room,” which I said was not a proper argument but just an intuition pump. I went on to say that intuition pumps are fine if they’re used correctly, but they can also be misused. They’re not arguments, they’re stories. Instead of having a conclusion, they pump an intuition. They get you to say “Aha! Oh, I get it!”

Further Reading:

Dennett, Daniel C. “The Intentional Stance”. 1989.

Dennett, Daniel C. “Consciousness Explained”. 1992.

The noteworthy Zentrum für virtual reality und visualisierung (Center for Virtual Reality and Visualization) in Vienna/Graz has a near-complete list of upcoming conferences in the field.

While the list is irreplacable in itself, the vrvis as an institution may also be worth a look. I do disagree with them on how valuable virtual reality really is, but they offer internships and the possibility to write diploma (PhDs probably as well) with them for up to 20 students.

Research areas include mainly medical and mathematical/physical visualizations.