Monday, 1 April 2013

Patterns; from tessellations to buying behaviour

A few weeks ago I was invited by the director of Museum Gouda, Gerard de Kleijn, to give a lecture. Not just a lecture on optimisation but one that would link an item from the museum to my profession. There are many ways in which art and math are related with the obvious topics being the golden ration, perspective, topology and fractals. Walking through the museum with Gerard, having a look at the stained glass windows, the ceramics, the art work from the Haagse School I realised that all had to do with modelling, the kind of think I do all the time using mathematics. The closest link to math I found in the various tessellations in the museum buildings which immediately gave me the theme of my lecture, Patterns. Math after all is also known as the science of patterns. See for example Hardy in A Mathematician’s Apology.

A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas
Patterns are very important to us humans. Our brains are pattern recognition machines because of our associative way of learning. Thousands of years ago these pattern recognition capabilities helped us survive because our brain could recognise the sound of a predator in the grass. A peculiar thing to note is that pattern recognition is a right brain activity, the part of the brain associated with creativity and design. The left side is associated with logic and analytical thinking. Does math modelling bring these two together?

In my lecture, starting from the tessellations in the museum buildings, I explained the math that goes into designing these kinds of tessellations. It’s basic basic geometry. A few hundred years back Sebastien Truchet (1657 –1729), a Dominican Father, was the first to study tessellation mathematically. He worked on the possible patterns one could make with square tiles that were divided diagonally. His model of pattern formation was later taken up by Fournier and is now known to mathematicians and designers as Truchet tiling. Fun to note maybe is that Truchet is also the designer of the font Romain le Roi, which we now know as Times New Roman (a designer’s worst nightmare?). Dutch designer M.C. Escher, famous for his mathematical inspired designs, took tessellations much further than Truchet. He took knowledge of many famous mathematicians like George PĆ³lya, Donald Coxeter and Roger Penrose and created wonderful woodcuts and lithographs. He became fascinated with tessellations after he had visited the Alhambra in Granada which is filled with mind blowing ancient tessellations. How those were created still is a puzzle. In Islamic tessellations not one but many types of tiles are combined, best-known are a set of tiles called the Girihtiles. Tessellations with these tiles go back to around 1200 and their arrangements found significant improvement starting with the Darb-i Imam shrine in Isfahan in Iran built in 1453. Today it is still not clear how the craftsmen at that time were able to create these wonderful and very complex tessellations as it is believed that the level of mathematical sophistication at that time was not high enough. It was only in 2007 that research showed this type of complex tessellations are comparable to Penrose tiling, predating those with five centuries.

Patterns not only appear in tessellations alone. Patterns can be found nearly everywhere. You see them in nature, in architecture, in our DNA, in fashion, in language, in music and in our buying behaviour. But instead of the clear patterns (and beauty) in tessellations, patterns in buying behaviour need to be discovered first, using advanced mathematical techniques. The basic material for pattern discovery in this case isn’t ceramic, stone, steel, or even glass but sales slips. Many grocery stores keep track of our purchases in brick and mortar stores and on line. Dutch firm Ahold for example registers over 4 billion individual items purchased each year. They have been doing so since 2000. Imagine the amount of data available to them. Until recently this data was used to develop efficient replenishment strategies for the stores and to optimise sourcing and stock levels.  Using advanced mathematical techniques like pattern recognition and machine learning they are now in search of our buying patterns.

Compared to grocery stores from 60 years ago, many things have changed. In those days the shop owner knew us well and was able to offer us suggestions that with high probability would fit our preferences. But in the 1950’s shops grew larger and instead of the grocer we collected the groceries we wanted ourselves. As a consequence knowledge of our preferences eroded, leaving the grocer (and the producers) clueless on what products to offer us. Today by keeping track of the items sold and using advanced mathematical techniques the grocer is regaining that knowledge. By mining for patterns in the vast amount of data the grocer can create a picture of his clientele; which groups of customers (tribes) visit his store, what defines them, what brands they buy and what their preferences are. Based on this information the grocer can better target these tribes in marketing, either in paper, online or even in-store, also called micro targeting. (see my blog)

Mining for patterns is not restricted to analysing buying behaviour. The techniques available have become incredibly powerful in a range of fields, from the workplace to the voting booth, from health care to counter-terrorism. However, great care must be taken when using the results from the data mining algorithms. First the algorithms search for correlations in the available data. That doesn’t mean that there actually is a causal relation. A famous example is that ice cream consumption causes drowning. As ice cream sales increase, so does the number of drowning deaths. The conclusion is however wrong because it fails to recognize the importance of temperature. As temperature rises, both ice consumption and water based activities increase (and therefore drowning deaths). Second, careful attention must be paid on the data used in the analysis. It is tempting to use the data that is available, that doesn’t imply that it is sufficient. It’s like the drunk looking for his keys under the lamppost and not in the dark alley where he lost them because it’s to dark to see in the alley. Under the lamp the light is much better. So think about what needs to be analysed and gather all relevant data. These are only two of many prerequisites for successful pattern discovery. Searching for patterns really is a specialist’s task, even if your statistical package supplier wants you to think different.

Math can be used to design complex and beautiful patterns; it is also required to discover patterns in mountains of unstructured data. Math indeed is the science of patterns. Applying is can create enormous value and beauty, but in the hands of untrained users it will be useless. To be effective training is required, becoming a numerical craftsman takes time and experience like the masters whose work is displayed in the Museum Gouda.