Models

There are things in this world that can be known with certainty. A good example: low counts. I have five fingers on each of my hands. No matter how often, when or where I count, the number is always five. No statistical distribution. Absolute certainty. (If we treat an established number system as a given, the only additional prerequisite is a good and stable definition of what constitutes a finger.)

Most other quantities come with some degree of uncertainty. Beginning with counts that extend to higher numbers (think of counting ballots in general elections), there is some counting error. (Note that strictly speaking, a wrong result in counting is not an error but a mistake. See the excellent article by Ben Buckner listed in the references below for details.) And if you need anything more sophisticated than adding up numbers, i. e. any measuring device, all bets are instantaneously off. With temperature, pressure, radiation, time, etc. you will always encounter some measurement error. You can minimise it, but it will never go away completely. You can never determine, say, a temperature with infinite precision. In other words, only an approximation of the real value is possible, an approximation of reality. And this is also known as a … model.

Descriptive Models

Singular or multiple measurements can constitute a descriptive model. E. g. the temperature in your refrigerator. If the display tells you 3.2 ° C, you don’t know the real value of (as this is fundamentally impossible, see above), but with some confidence you can assume that everything is fine.

Prescriptive Models

Keeping your mild in as safe temperature zone requires quite some fancy machinery. Not only the heat engine, which is a fascinating piece of kit in its own right, but also, crucially, a feedback mechanism, a loop of sensor, controller and actuator. Sensors and actuators are almost by definition dumb components. The controller does all the thinking. To hit the desired temperature range, it ought to have a rather clear picture of what the sensor data means and how the system (acted upon by the actuator) might respond. The controller needs a model.

Some predictive models are crude, e. g. a model that predicts the number of influenza cases during the next cold season. In more technical vernacular, such as model may be neither accurate (predicting the true value) nor precise (predicting a value with low variation). Other models are just that, highly accurate and precise, e. g. the exact time of day of sunrise (on a specific day and in a specific location). Or solar and lunar eclipses. In fact, celestial systems tend to behave extraordinarily predictable.

Dynamical Models

Dynamical models are models that evolve (change) over time. They are the interesting ones. By contrast, a static model is rather boring. (The book I am reading right now, ‘Escape from Model Land’ by Erica Thompson, weighs 185 g (with negligible variation due to ambient humidity). 185 g today, 185 g tomorrow. Nothing much happens. But this weight is still a model of the book, even though everything else, most of all the contents, is stripped away.) All things considered, we are, therefore, typically interested in dynamical models in order to make predictions. (Purely predictive models are only the default condition if making predictions is unattainable. A good example is seismology. The experts in this field know a whole lot about earthquakes. But predicting them is still an elusive goal that may not be reached for still a long time.)

Thinking about this, I have to refine the argument. Even static models can be, in fact, useful. Take Ohm’s law:

    \[I = \dfrac{V}{R}\]

Totally static (as there is no time parameter in it), but extremely predictive. For any given resistance R, a certain voltage V produces a defined current I. And this predictive nature is the crucial point. By contrast, the 185 g of Erica Thompson’s excellent book mentioned above aren’t able to predict anything (as long as we are not concerned with the energy expenditure required to hold it while reading; but let’s not overthink it).

Relationships

Often, building a model starts with observations. The more, the better (typically). Then patterns emerge, associations, relationships. In the end, the resulting model belongs to one of two categories:

  1. A (not so clear) statistical relationship. Ok, you can still fit a mathematical function (i. e. model) to the data, but you cannot explain the underlying mechanism completely. And the next incoming data point may even deviate widely from the prediction.
  2. A clear mathematical relationship, a function where a certain input value maps deterministically to a certain and precisely defined output value. As in Ohm’s law.

Speaking of mathematics. Take the following equation:

    \[sin \text{ } \theta = \dfrac{opposite}{hypotenuse}\]

Is this a model? My answer might surprise you. It is a definite ‘no’. Because this relationship between the angle and the sides of a right triangle is an incontrovertible mathematically proven fact. There is no uncertainty. This is not an approximation. This is not a model. This is truth.

Only in applying such a mathematical ‘law’ to real-world objects (physical or otherwise), it becomes a model in describing and predicting properties of objects and systems.

There might be a third category of models:

Algorithms: A function maps inputs to outputs. An algorithm does basically the same by applying an entire set of rules (or functions) to the input.

But there is a major constraint. If your algorithm (or function for that matter) is thought-out deterministically, it, by definition, produces truth and not an approximation. If the input is clearly defined, the output has no variation or distribution. In fact, algorithms and functions even resemble mathematical laws. Therefore, both are not models in and of themselves. They become models only through application to imprecise, noisy data. But if the input data is noise-free, the algorithm (or function) remains being not a model. It simply maps one truth to another.

    Further Reading

    Back Matter

    Copyright Thomas Gamsjäger

    Cite as: Gamsjäger T. Models. Maytensor 2026