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Head/tail breaks

Head/tail breaks is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Head/tail breaks rather than just read about it. In short: Head/tail breaks is a clustering algorithm for data with a heavy-tailed distribution such as power laws and lognormal distributions. The heavy-tailed distribution can be simply referred to the scaling pattern of far more small things than large ones, or alternatively numerous smallest, a very few largest, and some in between the smallest and largest.

Head/tail breaks — main illustration
Head/tail breaks — illustration

Key takeaways

  • Head/tail breaks belongs to science; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Head/tail breaks to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Head/tail breaks from memory before moving on to harder problems.

Reference excerpt

Head/tail breaks is a clustering algorithm for data with a heavy-tailed distribution such as power laws and lognormal distributions. The heavy-tailed distribution can be simply referred to the scaling pattern of far more small things than large ones, or alternatively numerous smallest, a very few largest, and some in between the smallest and largest. The classification is done through dividing things into large (or called the head) and small (or called the tail) things around the arithmetic mean or average, and then recursively going on for the division process for the large things or the head until the notion of far more small things than large ones is no longer valid, or with more or less similar things left only. Head/tail breaks is not just for classification, but also for visualization of big data by keeping the head, since the head is self-similar to the whole. Head/tail breaks can be applied not only to vector data such as points, lines and polygons, but also to raster data like digital elevation model (DEM).

Motivation The head/tail breaks is motivated by inability of conventional classification methods such as equal intervals, quantiles, geometric progressions, standard deviation, and natural breaks - commonly known as Jenks natural breaks optimization or k-means clustering to reveal the underlying scaling or living structure with the inherent hierarchy (or heterogeneity) characterized by the recurring notion of far more small things than large ones. Note that the notion of far more small things than large one is not only referred to geometric property, but also to topological and semantic properties. In this connection, the notion should be interpreted as far more unpopular (or less-connected) things than popular (or well-connected) ones, or far more meaningless things than meaningful ones. Head/tail breaks uses the mean or average to dichotomize a dataset into small and large values, rather than to characterize classes by average values, which is unlike k-means clustering or natural breaks. Through the head/tail breaks, a dataset is seen as a living structure with an inherent hierarchy with far more smalls than larges, or recursively perceived as the head of the head of the head and so on. It opens up new avenues of analyzing data from a holistic and organic point of view while considering different types of scales and scaling in spatial analysis.

Method Given some variable X that demonstrates a heavy-tailed distribution, there are far more small x than large ones. Take the average of all xi, and obtain the first mean m1. Then calculate the second mean for those xi greater than m1, and obtain m2. In the same recursive way, we can get m3 depending on whether the ending condition of no longer far more small x than large ones is met. For simplicity, we assume there are three means, m1, m2, and m3. This classification leads to four classes: [minimum, m1], (m1, m2], (m2, m3], (m3, maximum]. In general, it can be represented as a recursive function as follows:

Recursive function Head/tail Breaks: Rank the input data values from the biggest to the smallest; Compute the mean value of the data Break the data (around the mean) into the head and the tail; // the head for data values greater the mean // the tail for data values less the mean If (length(head)/length(data) <=40%): Head/tail Breaks(head); End Function

The resulting number of classes is referred to as ht-index, an alternative index to fractal dimension for characterizing complexity of fractals or geographic features: the higher the ht-index, the more complex the fractals.

Threshold or its sensitivity The criterion to stop the iterative classification process using the head/tail breaks method is that the remaining data (i.e., the head part) are not heavy-tailed, or simply, the head part is no longer a minority (i.e., the proportion of the head part is no longer less than a threshold such as 40%). This threshold is suggested to be 40% by Jiang et al. (2013), just as the codes above (i.e., (length/head)/length(data) ≤ 40%). This process is called head/tail breaks 1.0. But sometimes a larger threshold, for example 50% or more, can be used, as Jiang and Yin (2014) noted in another article: "this condition can be relaxed for many geographic features, such as 50 percent or even more". However, all heads' percentage on average must be smaller than 40% (or 41, 42%), indicating far more small things than large ones. Many real-world data cannot be fit into a perfect long tailed distribution, therefore its threshold can be relaxed structurally. In head/tail breaks 2.0 the threshold only applies to the overall heads' percentage. This means that the percentages of all heads related to the tails should be around 40% on average. Individual classes can have any percentage spit around the average, as long as this averages out as a whole. For example, if there is data distributed in such a way that it has a clearly defined head and tail during the first and second iteration (length(head)/(length(data)<20%) but a much less well defined long tailed distribution for the third iteration (60% in the head), head/tail breaks 2.0 allows the iteration to continue into the fourth iteration which can be distributed 30% head - 70% tail again and so on. As long as the overall threshold is not surpassed the head/tail breaks classification holds.

Rank-size plot and RA index A good tool to display the scaling pattern, or the heavy-tailed distribution, is the rank-size plot, which is a scatter plot to display a set of values according to their ranks. With this tool, a new index termed as the ratio of areas (RA) in a rank-size plot was defined to characterize the scaling pattern. The RA index has been successfully used in the estimation of traffic conditions. However, the RA index can only be used as a complementary method to the ht-index, because it is ineffective at capturing the scaling structure of geographic features.

Other Indices based on the head/tail breaks In addition to the ht-index, the following indices are also derived with the head/tail breaks.

… excerpt ends here. Continue reading the full article.

Illustrations

Head/tail breaks: 1024 cities that follow exactly Zipf's law, which implies that the first largest city is size 1, the second largest city is size 1/2, the third largest city is size 1/3, ... and the smallest city is size 1/1024. The left pattern is produced by head/tail breaks, while the right one by natural breaks, also known as Jenks natural breaks optimization.
1024 cities that follow exactly Zipf's law, which implies that the first largest city is size 1, the second largest city is size 1/2, the third largest city is size 1/3, ... and the smallest city is size 1/1024. The left pattern is produced by head/tail breaks, while the right one by natural breaks, also known as Jenks natural breaks optimization.
Head/tail breaks: An Illustration of the head/tail breaks classification with 10 numbers
An Illustration of the head/tail breaks classification with 10 numbers
Head/tail breaks: The left panel pattern contains 50,000 natural cities, which can be put into 7 hierarchical levels. It looks like a hair ball. Instead of showing all the 7 hierarchical levels, we show 4 top levels, by dropping out 3 low levels. Now with the right panel, the scaling pattern of far more small cities than large ones emerges. The right pattern (or the remaining part after dropping out the tails) is self-similar to the whole (or the left pattern). Thus the right pattern reflects the underlying structure of the left one, and enables us to see the whole.
The left panel pattern contains 50,000 natural cities, which can be put into 7 hierarchical levels. It looks like a hair ball. Instead of showing all the 7 hierarchical levels, we show 4 top levels, by dropping out 3 low levels. Now with the right panel, the scaling pattern of far more small cities than large ones emerges. The right pattern (or the remaining part after dropping out the tails) is self-similar to the whole (or the left pattern). Thus the right pattern reflects the underlying structure of the left one, and enables us to see the whole.
Head/tail breaks: The scaling pattern of US terrain surface is distorted by the natural breaks, but revealed by the head/tail breaks.
The scaling pattern of US terrain surface is distorted by the natural breaks, but revealed by the head/tail breaks.
Head/tail breaks: The spiral layout illustrates eight levels of hierarchy derived from a fictitious dataset of 5,000 cities, whose sizes follow a rank-size distribution. Cities are represented by points using a spectrum colormap, ranging from red (largest) to blue (smallest).
The spiral layout illustrates eight levels of hierarchy derived from a fictitious dataset of 5,000 cities, whose sizes follow a rank-size distribution. Cities are represented by points using a spectrum colormap, ranging from red (largest) to blue (smallest).

Worked examples

Example 1 — a first encounter with Head/tail breaks

Start with the simplest possible case. Write down what Head/tail breaks claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Head/tail breaks before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Head/tail breaks ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Head/tail breaks

In research
Head/tail breaks appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Head/tail breaks in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Head/tail breaks is common in secondary-school and first-year university syllabi. It links to neighbouring topics Big data, Data management, Geographic information systems, so understanding it makes those chapters shorter.
In everyday life
Look for Head/tail breaks outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.
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How to study Head/tail breaks in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Head/tail breaks means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Head/tail breaks out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Head/tail breaks in simple terms?

Head/tail breaks is a clustering algorithm for data with a heavy-tailed distribution such as power laws and lognormal distributions. The heavy-tailed distribution can be simply referred to the scaling pattern of far more small things than large ones, or alternatively numerous smallest, a very few l…

Why does Head/tail breaks matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Head/tail breaks?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Head/tail breaks.

Tags

  • Big data
  • Data management
  • Geographic information systems

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