The Art of Dimensionalisation
The Art of Dimensionalisation← you are here
Science progresses by naming what matters. Not just measuring more, or modelling faster, but finding the right dimensions to describe a problem, such that explanation, prediction, and intervention become tractable. This act, dimensionalisation, sits at the core of theory formation. Discovery of new dimensions enables new understandings, it is the key to creativity.
To dimensionalise (or parameterise) is to propose that a phenomenon varies along some axis worth tracking, that differences in outcome follow from differences in dimension, not just from noise or context. It is how we convert vague puzzles into workable frames. It is also how we make theories portable.
Why dimensionalisation is epistemic, not technical
In statistical or computational work, we often treat parameters as numbers to be estimated. But before anything is estimated, it is first named. What counts as a “parameter” in a theory is a decision: it tells us what we expect variation in the world to depend on. For instance:
In physics: mass, charge, momentum
In social science: trust, risk, identity
In transport: time, cost, reliability, access
Each is a construct, but also a dimension: something with scope, variation, and presumed causal force. Dimensionalisation names and bounds such constructs in a way that opens up structured investigation.
From Insight to Prediction: The Arc of Theory Formation
The journey from new idea to useful theory follows a repeatable arc: from dimensional insight, to construct definition, to parameterised prediction.
1. Dimensional Insight
All theory starts with noticing. You fit a model, and the residuals are structured. Two cases that should behave similarly, don’t. A variable that mattered before suddenly fails. A paradox appears, a regularity breaks.
You suspect the model is missing something, not just noise, but a dimension. A latent axis of variation that, if named, might explain what your current frame cannot.
Insight begins qualitatively:
People don’t just care about distance, they care about time of day.
Two locations are equally accessible, but one feels less reliable.
Urban density doesn’t explain the difference, but synchrony might.
2. Parameterising the Dimension
Naming a dimension is the first act of theory. Parameterising it is the second.
This means:
Defining what the dimension means, and when it matters.
Constructing a measurable proxy.
Choosing a unit, a scale, and a sign.
Deciding how it enters the model: additively, interactively, or as a threshold.
Example: You propose synchrony, the overlap of the traveller’s window, the destination’s open hours, and the transit service span (hours of operation), as a missing dimension in accessibility models. It is quantified:
[latex]s_{ij} = \frac{|W_i \cap A_j \cap S_{ij}|}{|W_i|}[/latex]
You incorporate this into a cumulative-opportunity model:
[latex]A_i = \sum_j t_{ij} \le T] \cdot s_{ij} \cdot O_j[/latex]
Now synchrony is not just a hunch, it is a parameterised construct, ready for empirical test.
3. Making Predictions
With a parameterised model, you can now ask: what changes when this dimension changes?
That means:
Writing claims: “If synchrony increases, access rises.”
Stating interventions: “Span extensions should disproportionately benefit night workers.”
Testing subgroup predictions and out-of-sample performance.
A good parameter is one that:
Explains variation the model previously missed.
Makes novel, testable predictions.
Improves decision relevance in real contexts.
How dimensionalisation builds theory
There are (at least) four roles parameterisation plays in theory development:
1. Dimensional declaration
“This matters. Its variation helps explain what we see.”
Choosing parameters is a commitment to a dimensional story. When we say that travel behaviour depends on time and cost, or that access depends on speed and synchrony, we are declaring what kinds of difference make a difference.
2. Construct organisation
“These things cluster or conflict.”
Dimensionalisation lets us represent tradeoffs (e.g. time vs. reliability), composite effects (e.g. generalised cost), or separable dimensions (e.g. geographic vs. temporal access). It gives structure to our model space.
3. Comparative portability
“The same dimension appears here, too.”
A good parameter can travel. Synchrony, for example, links not just commuting and shift work, but also clinic appointment systems, freight scheduling, and school bell times. A parameter that unifies these is a new conceptual hinge.
4. Theory refinement
“This parameter no longer fits—replace, reframe, restrict.”
When a parameter fails (e.g. doesn’t generalise, leads to paradoxes, obscures rather than clarifies), theory changes by reorganising its dimensions. Progress does not always proceed by adding more parameters, it is often by choosing fewer, better ones. Less is more.
How to discover a new dimension
A practical sequence looks like this:
Dimensionalise explicitly: start by writing down the dimensions your current model uses (and what they leave out).
Collect structured failures: find residuals or cases where current dimensions don’t explain differences.
Propose a new dimension: what axis might explain the residual pattern? Name it.
Test portability: does this construct appear (even informally) in related domains?
Redimensionalise minimally: keep other dimensions fixed, and add the new one parsimoniously.
Refit and test: does it improve fit, explanation, or decision relevance?
Stress-test across contexts: does it hold out of sample? Is it useful beyond the training frame?
This is not just exploratory regression. It’s the systematic reframing of a problem by renaming what varies.
Synchrony, Revisited
In access models, traditional parameters like distance or speed sometimes fail to explain differences, particularly for night-shift workers, caregivers, or those with irregular schedules.
Synchrony becomes a promising new dimension:
It is measurable (as time-window overlap).
It generalises across domains (retail, freight, healthcare).
It improves prediction where old models fail.
It reflects that availability is relational, not absolute.
Dimensionalisation in Theory Graphs
A well-formulated dimension becomes a construct node in a Theory Graph:
It has a definition.
It links to one or more operational measures.
It is used by specific model variants.
It enables claims, which are tested by evidence.
It can be replaced, constrained, or reframed.
Changing a dimension reshapes this subgraph. By making these moves visible, theory formation becomes inspectable, revisable, and reusable.
Why this matters
Without dimensionalisation, models become black boxes and theories become brittle.
With good dimensionalisation, we can:
Explain anomalies.
Generalise across domains.
Compare theories with shared parts.
Improve decisions by reframing what matters.
This is not about adding complexity. It is about choosing the right axes, so that variation in the world is matched by variation in our explanations.
In short
Dimensionalisation is the scientist’s real superpower.
It is how we say: this difference matters, and this frame will hold.
And it is how we build theories that do not merely fit the data, but actually help us think.
