Modelling: The Red Cherry Problem

Modelling (noun) /ˈmɒd(ə)lɪŋ/

1. The construction of a simplified representation of a system or process, often involving mathematical relationships, intended to describe, predict, or understand real-world phenomena.
2. (Informal, sardonic) The art of making assumptions you can’t justify, using parameters you can’t measure, to generate predictions you can’t test - typically in support of decisions already taken.

Usage note: Modelling is frequently invoked post hoc to lend a veneer of scientific legitimacy to actions rooted in instinct, politics, or panic. While often presented with graphs, equations, and confidence intervals, the robustness of a model depends less on its complexity than on the clarity of its assumptions and the quality of its data - both of which are usually in short supply.

Modelling rose to prominence during the twentieth century. But it has taken a bit of a beating in recent years. Increasingly politicized it’s almost a dirty word. You can thank pandemic forecasts, climate projections, and economic predictions for that. Models promised clarity, but when they tell us things we don't want to hear, then they become a political football. There for everyone to kick around. A polite way of saying we’re making it up as we go along. Fancy guesses dressed in Greek. Projections in PowerPoint. Magick spells cast in code.

But there's nothing wrong with modelling.

Humans have been modelling for millions of years. We just didn’t call it that.

Humans are natural-born modellers. But terrible statisticians.

The Red Cherry Problem

Modelling is intuitive. Kids do it all the time.

Don't believe me?

Take a look at the four panels below.

They show the value of a variable, say Yield, as a function of two variables - T and B - plotted for values of A=10, 20, 30 and 40.

The Yield values range from about 5ug to 80ug.

The values are represented as squares of different sizes and colours.

The small green squares are close to the minimum.

The large red squares are the maximum values.

By representing the Yield values as coloured squares we are emulating an ecologically valid task. Instead of an abstract representation we are framing the problem as a foraging task.

This is The Red Cherry Problem.

The small green cherries are inedible.

The LARGE RED cherries are ripe.

Notice that each panel has one value missing?

But intuitive modelling let's you fill in the blanks.

Given a choice of squares of different colours and different sizes, you should be able to complete the missing data cells for all four of the graphs in less than 20s.

How big and how red are the values in the missing squares?

Here's the solution:

We can do this easily, because we are good at recognizing patterns and comfortable interpolating data.

Design, Structure, Balance

In addition, by laying the results out systematically in this way, we discover that we can quickly work out what is going on in this system.

• When A is low (Panel A=10) changing B and T has pretty much no discernible effect.
• But, as A increases, through A=20 and A=30, the cherries get bigger and riper.
• Beyond 30 we see little improvement. This suggests there is curvature: as A increases we see improvements until we hit some kind of upper boundary. The cherries are no bigger or better at A=40 than they are at A=30.
• And, as we increase B, the cherries get bigger. But these improvements are best at higher temperatures, T. There is a BxT interaction.

No modelling, nothing fancy, the structure becomes obvious once the data are laid out systematically.

This structure is the essence of experimental design DOE tools. DOE tools select design points to simplify interpretation. Lay the data out systematically and you are half way there. You don't need a fancy analysis to understand what is happening.

If your experiment needs statistics, you ought to have done a better experiment - Ernest Rutherford

Note: This quote was later hijacked and branded as their own by Statistics-Deniers.

A well-designed experiment should yield clear, unambiguous results without relying heavily on statistical analysis to draw conclusions. It implies that if your experiment requires complex statistical methods to tease out a meaningful result, the experiment itself might not be terribly robust or well-controlled.

The Limits of Intuition

Humans are really pretty good at pattern recognition. And we are really good at interpolating data. We had to be.

Get this wrong during the Pleistocene and the chances were you'd find a sabre-toothed tiger sitting on your chest with your windpipe in its jaws.

Those that weren't good at pattern recognition never lived to tell the tale.

We evolved powerful pattern recognition making Homo sapiens an intuitive modeller.

Kids have it from an early age.

They can interpolate with ease in two, three, even four dimensions. More than that and we start to struggle.

The Pleistocene left us with remarkably good intuitive modelling instincts for low dimensional spaces. We evolved to track trajectories, assess threats, and optimize foraging routes in 3D space, often using nothing more than visual cues, gut feeling, and rules of thumb. Our ancestors didn’t need matrix algebra to forage for ripe berries or evade predators. They needed spatial memory, pattern recognition, and the ability to generalise from experience.

And when we cast a modelling problem in ecologically realistic terms - say, foraging for berries - we tap into these ancient skills. We invoke a form of natural modelling that feels effortless and instinctive. It's child's play.

Games such as Spot the Difference exploit pattern recognition - anomaly detection within a visual field was important to us.

The problem comes when we stray beyond those dimensions. Once we move into 5, 10, or n-dimensional spaces our intuitive skills collapse. We fall off the edge of the cognitive map.

At that point, we must resort to matrix algebra.

While the Pleistocene left us with good pattern recognition, our statistical intuition never evolved. We’re wired for stories, not significance. For shape recognition, not sampling error.

And this is why statistical tools like the Design of Experiments (DOE) are so essential.

Because the tools of modern science aren’t instinctive.

And while our evolutionary toolkit makes us feel confident in our interpretation of data, statistical tools are desperately seeking to compensate for the blind spots left by evolution.

Those tools served us well.

Evolution favoured fast heuristics, rules of thumb, intuition over computation.

But this isn't the Pleistocene.

This is the Holocene.

This is the Space-Age.

Born Modellers: In the Pleistocene, getting it wrong meant getting eaten.